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{"source/main.ptx": ["my-great-book"], "source/frontmatter.ptx": ["frontmatter", "front-colophon"], "source/ch-Probability.ptx": ["ch-Probability", "sec-Set-Theory", "sec-Probability", "sec-Conditional-Probability", "sec-Independent-Events"], "source/ch-Random-Variables.ptx": ["ch-Random-Variables", "sec-Discrete-RVs", "sec-Continuous-RVs"], "source/ch-Expected-Value.ptx": ["ch-Expected-Value", "sec-Expected-Value", "sec-Variance", "sec-Covariance"], "source/ch-Confidence-Intervals.ptx": ["ch-Confidence-Intervals", "sec-CLT", "sec-Confidence-Intervals"], "source/ch-Hypothesis-Testing.ptx": ["ch-Hypothesis-Testing", "sec-One-Sample-Tests", "sec-Two-Sample-Tests", "sec-Power", "sec-Chi-Squared"], "source/ch-Linear-Regression.ptx": ["ch-Linear-Regression", "sec-Correlation", "sec-Linear-Regression"], "source/backmatter.ptx": ["backmatter"]}
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<meta property="og:type" content="book">
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<meta property="book:title" content="Math 1044 Notes">
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<meta property="book:author" content="You">
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<meta property="book:author" content="Andy Eisenberg">
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<script src="_static/pretext/js/lib/jquery.min.js"></script><script src="_static/pretext/js/lib/jquery.sticky.js"></script><script src="_static/pretext/js/lib/jquery.espy.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></script><script src="_static/pretext/js/user_preferences.js"></script><!--** eBookCongig is necessary to configure interactive **-->
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<body id="changeme" class="pretext book ignore-math">
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<body class="pretext book ignore-math">
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<a class="assistive" href="#ptx-content">Skip to main content</a><header id="ptx-masthead" class="ptx-masthead"><div class="ptx-banner"><div class="title-container">
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||||
<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<p class="byline">You</p>
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<p class="byline">Andy Eisenberg</p>
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||||
</div></div></header><nav id="ptx-navbar" class="ptx-navbar navbar"><div class="ptx-navbar-contents">
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<button class="toc-toggle button" title="Contents"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Contents</span></button><div class="searchbox">
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<div class="searchwidget"><button id="searchbutton" class="searchbutton button" type="button" title="Search book"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Search Book</span></button></div>
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@@ -130,7 +130,20 @@ eBookConfig.enable_chatcodes = false;
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<span class="nav-other-controls"><button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="backmatter.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="backmatter.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><span class="next-button button disabled"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></span></span>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R}
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
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@@ -142,13 +155,99 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
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||||
<li class="toc-item toc-chapter">
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||||
<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-section-name.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Section Title</span></a></div></li></ul>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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</ul>
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</li>
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<li class="toc-item toc-chapter">
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||||
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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||||
</ul>
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||||
</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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||||
<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
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||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
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||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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</ul>
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</li>
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div></li>
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div></li>
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div></li>
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div></li>
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div></li>
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<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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</ul></nav></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="colophon" id="backmatter-2"><h2 class="heading hide-type">
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<meta name="viewport" content="width=device-width, initial-scale=1.0">
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<meta property="og:type" content="book">
|
||||
<meta property="book:title" content="Math 1044 Notes">
|
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<meta property="book:author" content="You">
|
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<meta property="book:author" content="Andy Eisenberg">
|
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<script src="_static/pretext/js/lib/jquery.min.js"></script><script src="_static/pretext/js/lib/jquery.sticky.js"></script><script src="_static/pretext/js/lib/jquery.espy.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></script><script src="_static/pretext/js/user_preferences.js"></script><!--** eBookCongig is necessary to configure interactive **-->
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@@ -113,10 +113,10 @@ eBookConfig.enable_chatcodes = false;
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<body id="changeme" class="pretext book ignore-math">
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<body class="pretext book ignore-math">
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<a class="assistive" href="#ptx-content">Skip to main content</a><header id="ptx-masthead" class="ptx-masthead"><div class="ptx-banner"><div class="title-container">
|
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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<p class="byline">You</p>
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<p class="byline">Andy Eisenberg</p>
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|
||||
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|
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|
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-section-name.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Section Title</span></a></div></li></ul>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
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|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
|
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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|
||||
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|
||||
<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
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|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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|
||||
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|
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|
||||
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|
||||
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
|
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</ul>
|
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</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
|
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|
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|
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div></li>
|
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<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div></li>
|
||||
<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div></li>
|
||||
<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div></li>
|
||||
<li class="toc-item toc-chapter"><div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div></li>
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</ul></nav></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="frontmatter" id="frontmatter"><h2 class="heading"><span class="title">Math 1044 Notes</span></h2>
|
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<div class="author">
|
||||
<div class="author-name">You</div>
|
||||
<div class="author-info">Your department<br>Your institution</div>
|
||||
<div class="author-name">Andy Eisenberg</div>
|
||||
<div class="author-info">Mathematics Department<br>Temple University</div>
|
||||
</div>
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<div class="date">December 28, 2025</div>
|
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<div class="date">December 31, 2025</div>
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para">Let <span class="process-math">\(X\)</span> take values in the interval <span class="process-math">\([a, b]\text{.}\)</span> We say that <span class="process-math">\(X\)</span> has the <dfn class="terminology">uniform distribution</dfn> if, for any subinterval <span class="process-math">\([c, d] \subset [a, b]\text{,}\)</span> the probability that <span class="process-math">\(X\)</span> lies within <span class="process-math">\([c, d]\)</span> is equal to the proportion of the total sample space taken by <span class="process-math">\([c, d]\text{.}\)</span> That is:</div>
|
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<div class="displaymath process-math">
|
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\begin{gather*}
|
||||
\Pr(c \leq X \leq d) = \frac{d - c}{b - a}.
|
||||
\end{gather*}
|
||||
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|
||||
<div class="para">In this case, <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span>
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<div class="autopermalink" data-description="Definition 2.2.2"><a href="#def-continuous-uniform-distribution" title="Copy heading and permalink for Definition 2.2.2" aria-label="Copy heading and permalink for Definition 2.2.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-continuous-uniform-distribution">in-context</a></span>
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
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|
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<div class="para logical">
|
||||
<div class="para">Suppose we observe traffic along a particular stretch of highway which typically has 200 cars pass per hour. Let <span class="process-math">\(N\)</span> be the number of cars seen in a 2-hour observation period. Since the parameter <span class="process-math">\(\lambda = 200\)</span> cars per hour is given for a 1-hour period, we should scale it to find the rate for a 2-hour period: <span class="process-math">\(\lambda t = (200)(2) = 400\)</span> cars per 2-hours. Then, for example:</div>
|
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<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
\Pr(N = 375) = p(375; 200, 2) = \frac{400^{375}}{375!} e^{-400} \approx 0.01.
|
||||
\end{gather*}
|
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</div>
|
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<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div> <div class="para logical">
|
||||
<div class="para">Now consider the random variable <span class="process-math">\(M\)</span> counting the number of cars seen in a 30-minute observation period. We should not calculate <span class="process-math">\(\lambda t = (200)(30) = 6000\text{,}\)</span> since <span class="process-math">\(\lambda\)</span> measures time in hours and <span class="process-math">\(t\)</span> measures time in minutes. We should first convert <span class="process-math">\(t = 0.5\)</span> hours, then we can find our appropriately scaled rate information: <span class="process-math">\((200)(0.5) = 100\)</span> cars per half-hour. Then, for example:</div>
|
||||
<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
\Pr(M = 110) = p(110; 200, 0.5) = \frac{100^{110}}{110!}e^{-100} \approx 0.02.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Note that when we write the notation <span class="process-math">\(p(k; \lambda, t)\text{,}\)</span> we assume that <span class="process-math">\(\lambda\)</span> and <span class="process-math">\(t\)</span> are expressed with the same time units already.</div>
|
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<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
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<div class="autopermalink" data-description="Example 2.1.17"><a href="#example-poisson-rate" title="Copy heading and permalink for Example 2.1.17" aria-label="Copy heading and permalink for Example 2.1.17">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#example-poisson-rate">in-context</a></span>
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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|
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<div class="para logical">
|
||||
<div class="para">An experiment consists of rolling a fair 6-sided die two times. Let <span class="process-math">\(A\)</span> be the event that the sum of the rolls is at least 10. To find <span class="process-math">\(\Pr(A)\text{,}\)</span> we note that <span class="process-math">\(A = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\}\text{,}\)</span> which has 6 elements. Since <span class="process-math">\(|\Omega| = 36\text{,}\)</span> we have</div>
|
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||||
\begin{gather*}
|
||||
\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{36} = \frac{1}{6}\text{.}
|
||||
\end{gather*}
|
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|
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<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <div class="para logical">
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||||
<div class="para">Let <span class="process-math">\(B\)</span> be the event that the first roll is 6. If we pause after the first die roll seeing the value of 6, we might be more inclined to expect a sum of at least 10. The evidence that we’ve already seen changes our understanding of the situation. Since <span class="process-math">\(|B| = 6\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Also, <span class="process-math">\(A\cap B = \{(6, 4), (6, 5), (6, 6)\}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = \frac{3}{36} = \frac{1}{12}\text{.}\)</span> Finally:</div>
|
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<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{1/12}{1/6} = \frac{1}{2}
|
||||
\end{gather*}
|
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<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 1.3.2"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 1.3.2" aria-label="Copy heading and permalink for Example 1.3.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Conditional-Probability.html#example-rolls-conditional">in-context</a></span>
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||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.12</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Binomial Probability Formula.</span>
|
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|
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|
||||
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|
||||
\begin{gather*}
|
||||
b(k; n, p) = {n \choose k} p^k (1-p)^{n-k}
|
||||
\end{gather*}
|
||||
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<div class="autopermalink" data-description="Paragraph"><a href="#fact-binomial-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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|
||||
<div class="autopermalink" data-description="Fact 2.1.12: Binomial Probability Formula"><a href="#fact-binomial-probability-formula" title="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#fact-binomial-probability-formula">in-context</a></span>
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<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Bin(3, 0.4)\)</span><div class="autopermalink" data-description="Table 2.1.8: \Bin(3, 0.4)"><a href="#table-binomial-example" title="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)" aria-label="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
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|
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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
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|
||||
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|
||||
<td class="c m b0 r0 l0 t0 lines">0</td>
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<td class="c m b0 r0 l0 t0 lines">2</td>
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<tr>
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||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.064</td>
|
||||
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||||
</table></div></figure><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#table-binomial-example">in-context</a></span>
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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<p class="byline">You</p>
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<p class="byline">Andy Eisenberg</p>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R}
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<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
|
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-section-name.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Section Title</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
|
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
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|
||||
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|
||||
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
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||||
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||||
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|
||||
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
|
||||
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|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
|
||||
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||||
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||||
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||||
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|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
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|
||||
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|
||||
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|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
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|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
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|
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||||
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|
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|
||||
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|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
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<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2 focused" data-preexpanded-levels="0" data-max-levels="2"><ul class="structural toc-item-list contains-active">
|
||||
<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
|
||||
<ul class="structural toc-item-list contains-active">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
|
||||
<ul class="structural toc-item-list active">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
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</li>
|
||||
</ul>
|
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||||
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
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</li>
|
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</ul>
|
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
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</ul>
|
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<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
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<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
|
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|
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
|
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</ul>
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<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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</ul></nav></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Conditional-Probability"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Conditional Probability</span>
|
||||
</h2>
|
||||
<section class="subsection" id="subsec-conditional-probability"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Conditional Probability</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="def-conditional-probability"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-conditional-probability-1-1">
|
||||
<div class="para">Let <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> be events. The <dfn class="terminology">conditional probability</dfn> of <span class="process-math">\(A\)</span> given <span class="process-math">\(B\)</span> is</div>
|
||||
<div class="displaymath process-math" id="def-conditional-probability-1-1-6">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-conditional-probability-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.3.1"><a href="#def-conditional-probability" title="Copy heading and permalink for Definition 1.3.1" aria-label="Copy heading and permalink for Definition 1.3.1">🔗</a></div></article><article class="example example-like" id="example-rolls-conditional"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="example-rolls-conditional-1-1">
|
||||
<div class="para">An experiment consists of rolling a fair 6-sided die two times. Let <span class="process-math">\(A\)</span> be the event that the sum of the rolls is at least 10. To find <span class="process-math">\(\Pr(A)\text{,}\)</span> we note that <span class="process-math">\(A = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\}\text{,}\)</span> which has 6 elements. Since <span class="process-math">\(|\Omega| = 36\text{,}\)</span> we have</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-1-5">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{36} = \frac{1}{6}\text{.}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="example-rolls-conditional-1-2">
|
||||
<div class="para">Let <span class="process-math">\(B\)</span> be the event that the first roll is 6. If we pause after the first die roll seeing the value of 6, we might be more inclined to expect a sum of at least 10. The evidence that we’ve already seen changes our understanding of the situation. Since <span class="process-math">\(|B| = 6\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Also, <span class="process-math">\(A\cap B = \{(6, 4), (6, 5), (6, 6)\}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = \frac{3}{36} = \frac{1}{12}\text{.}\)</span> Finally:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-conditional-1-2-6">
|
||||
\begin{gather*}
|
||||
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{1/12}{1/6} = \frac{1}{2}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 1.3.2"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 1.3.2" aria-label="Copy heading and permalink for Example 1.3.2">🔗</a></div></article><div class="autopermalink" data-description="Subsection 1.3.1: Conditional Probability"><a href="#subsec-conditional-probability" title="Copy heading and permalink for Subsection 1.3.1: Conditional Probability" aria-label="Copy heading and permalink for Subsection 1.3.1: Conditional Probability">🔗</a></div></section><section class="subsection" id="subsec-diagnostic-testing"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Diagnostic Testing</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-diagnostic-testing-2">Diagnostic tests for diseases aren’t perfect. When a test comes back positive or negative, a patient will want to understand the (conditional) probability that they have or don’t have the disease based on the evidence (the diagnostic test result).<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-sensitivity-specificity"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-sensitivity-specificity-1-1">The <dfn class="terminology">sensitivity</dfn> of a diagnostic test is the probability that a patient who has the disease will see a positive test result. The <dfn class="terminology">specificity</dfn> is the probability that a patient who does not have the disease will see a negative test result.<div class="autopermalink" data-description="Paragraph"><a href="#def-sensitivity-specificity-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.3.3"><a href="#def-sensitivity-specificity" title="Copy heading and permalink for Definition 1.3.3" aria-label="Copy heading and permalink for Definition 1.3.3">🔗</a></div></article><div class="para" id="subsec-diagnostic-testing-4">Introducing event notation, let <span class="process-math">\(D\)</span> be the event that a patient has the disease, and let <span class="process-math">\(P\)</span> be the event that they receive a positive test result. Then the sensitivity of the diagnostic test is <span class="process-math">\(\Pr(P \mid D)\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c \mid D^c)\text{.}\)</span> However, when the patient takes a diagnostic test, the conditional probabilities they would be most interested in would be <span class="process-math">\(\Pr(D \mid P)\)</span> and <span class="process-math">\(\Pr(D^c \mid P^c)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-diagnostic-testing-5">Bayes’ Theorem expresses the relationship between a conditional probability <span class="process-math">\(\Pr(A \mid B)\)</span> and the flipped conditional probability <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="theorem theorem-like" id="thm-Bayes-v1"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v1).</span>
|
||||
</h4>
|
||||
<div class="para logical" id="thm-Bayes-v1-2-1">
|
||||
<div class="displaymath process-math" id="thm-Bayes-v1-2-1-1">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<div class="autopermalink" data-description="Theorem 1.3.4: Bayes’ Theorem (v1)"><a href="#thm-Bayes-v1" title="Copy heading and permalink for Theorem 1.3.4: Bayes’ Theorem (v1)" aria-label="Copy heading and permalink for Theorem 1.3.4: Bayes’ Theorem (v1)">🔗</a></div></article><div class="para logical" id="subsec-diagnostic-testing-7">
|
||||
<div class="para">For example, if a patient sees a positive diagnostic test result, they might try to calculate:</div>
|
||||
<div class="displaymath process-math" id="subsec-diagnostic-testing-7-1">
|
||||
\begin{gather*}
|
||||
\Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">It will take some work to be able to use this formula. We likely don’t have direct access to <span class="process-math">\(\Pr(P)\)</span> or <span class="process-math">\(\Pr(D)\text{.}\)</span>
|
||||
</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="para logical" id="subsec-diagnostic-testing-8">
|
||||
<div class="para">Let’s consider <span class="process-math">\(\Pr(P)\text{,}\)</span> the probability of receiving a positive test result. The sensitivity <span class="process-math">\(\Pr(P \mid D)\)</span> tells us this probability under the condition that the patient has the disease. For a patient who doesn’t have the disease, the specificity isn’t quite the number we’re looking for. However, consider the complementary probability:</div>
|
||||
<div class="displaymath process-math" id="subsec-diagnostic-testing-8-3">
|
||||
\begin{gather*}
|
||||
\Pr(P \mid D^c) = 1 - \Pr(P^c \mid D^c)
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">The total <span class="process-math">\(\Pr(P)\)</span> can be divided into two categories: patients who have the disease and test positive, and patients who don’t have the disease and test positive. So:</div>
|
||||
<div class="displaymath process-math" id="subsec-diagnostic-testing-8-5">
|
||||
\begin{align}
|
||||
\Pr(P) \amp = \Pr(P\cap D) + \Pr(P\cap D^c) \tag{1.3.1}\\
|
||||
\amp = \Pr(P\mid D)\Pr(D) + \Pr(P \mid D^c)\Pr(D^c) \tag{1.3.2}\\
|
||||
\amp = \Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c \mid D^c))\Pr(D^c) \tag{1.3.3}\\
|
||||
\amp = (\text{sensitivity})\Pr(D) + (1 - \text{ specificity})\Pr(D^c) \tag{1.3.4}
|
||||
\end{align}
|
||||
</div>
|
||||
<div class="para">We can take this breakdown of <span class="process-math">\(\Pr(P)\)</span> and write a new version of Bayes’ Theorem:</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="theorem theorem-like" id="thm-Bayes-v2"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Bayes’ Theorem (v2).</span>
|
||||
</h4>
|
||||
<div class="para logical" id="thm-Bayes-v2-2-1">
|
||||
<div class="displaymath process-math" id="thm-Bayes-v2-2-1-1">
|
||||
\begin{gather*}
|
||||
\Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A \mid B)\Pr(B) + (1 - \Pr(A^c \mid B^c))\Pr(B^c)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Bayes-v2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 1.3.5: Bayes’ Theorem (v2)"><a href="#thm-Bayes-v2" title="Copy heading and permalink for Theorem 1.3.5: Bayes’ Theorem (v2)" aria-label="Copy heading and permalink for Theorem 1.3.5: Bayes’ Theorem (v2)">🔗</a></div></article><div class="para" id="subsec-diagnostic-testing-10">We’re still missing a crucial piece of information: <span class="process-math">\(\Pr(D)\text{,}\)</span> the probability (not conditioned on any evidence) that the patient has the disease. This is often referred to as the <dfn class="terminology">prior</dfn>, as in, our prior understanding of the probability of something before we gained some new information from evidence. The conditional probability calculated using Bayes’ Theorem is usually called the <dfn class="terminology">posterior</dfn> (i.e., after taking evidence into account). There isn’t always one single number that’s reasonable to use as the prior probability. For example, in a diagnostic testing situation, the <dfn class="terminology">prevalence</dfn> of the disease—i.e., the proportion of the population who have the disease—might feel like a natural number to use as the prior. However, what prevalence should you use? During the COVID-19 pandemic, the prevalence of COVID in a particular country, state, and city might be different. There’s also the possibility of applying Bayes’ Theorem multiple times to take into account multiple pieces of evidence, using the posterior probability from one application of Bayes’ Theorem to play the role of the prior probability in the next. This idea would apply if, for example, a patient took a second diagnostic test to double-check.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsec-diagnostic-testing-11"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-diagnostic-testing-11-1-1">A 50-year old woman with no symptoms is screened for breast cancer and tests positive. If the prevalence of breast cancer for women in her age group is 1% and the particular screening process used has a sensitivity of 90% and a specificity of 91%, what is the probability that the woman has breast cancer given her positive result?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-11-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="subsec-diagnostic-testing-11-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="subsec-diagnostic-testing-11-2-1">
|
||||
<div class="para">Let <span class="process-math">\(P\)</span> be the event of testing positive and <span class="process-math">\(D\)</span> the event of having the disease. Then the prevalence <span class="process-math">\(\Pr(D)\)</span> is given as 1%, or 0.01. The sensitivity is <span class="process-math">\(\Pr(P\mid D) = 0.9\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c\mid D^c) = 0.91\text{.}\)</span> So, according to Bayes’ Theorem:</div>
|
||||
<div class="displaymath process-math" id="subsec-diagnostic-testing-11-2-1-6">
|
||||
\begin{align*}
|
||||
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\mid D^c))\Pr(D^c)} \\
|
||||
\amp = \frac{(0.9)(0.01)}{(0.9)(0.01) + (1 - 0.91)(1 - 0.01)} \\
|
||||
\amp \approx 0.092
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-diagnostic-testing-11-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.3.6.1"><a href="#subsec-diagnostic-testing-11-2" title="Copy heading and permalink for Solution 1.3.6.1" aria-label="Copy heading and permalink for Solution 1.3.6.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Example 1.3.6"><a href="#subsec-diagnostic-testing-11" title="Copy heading and permalink for Example 1.3.6" aria-label="Copy heading and permalink for Example 1.3.6">🔗</a></div></article><div class="autopermalink" data-description="Subsection 1.3.2: Diagnostic Testing"><a href="#subsec-diagnostic-testing" title="Copy heading and permalink for Subsection 1.3.2: Diagnostic Testing" aria-label="Copy heading and permalink for Subsection 1.3.2: Diagnostic Testing">🔗</a></div></section><section class="exercises" id="exercises-Conditional-Probability"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<div class="exercisegroup" id="exercises-Conditional-Probability-1">
|
||||
<h4 class="heading"><span class="title">Exercise Group.</span></h4>
|
||||
<div class="introduction" id="exercises-Conditional-Probability-1-1"><div class="para" id="exercises-Conditional-Probability-1-1-1">In each of the following scenarios with given events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> alculate <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> <span class="process-math">\(\Pr(A\cap B)\text{,}\)</span> <span class="process-math">\(\Pr(A \mid B)\text{,}\)</span> and <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div>
|
||||
<div class="exercisegroup-exercises">
|
||||
<article class="exercise exercise-like" id="exercises-Conditional-Probability-1-2"><h5 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
|
||||
<div class="para" id="exercises-Conditional-Probability-1-2-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.3.3.1"><a href="#exercises-Conditional-Probability-1-2" title="Copy heading and permalink for Exercise 1.3.3.1" aria-label="Copy heading and permalink for Exercise 1.3.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Conditional-Probability-1-3"><h5 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
|
||||
<div class="para" id="exercises-Conditional-Probability-1-3-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
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|
||||
<div class="autopermalink" data-description="Exercise 1.3.3.2"><a href="#exercises-Conditional-Probability-1-3" title="Copy heading and permalink for Exercise 1.3.3.2" aria-label="Copy heading and permalink for Exercise 1.3.3.2">🔗</a></div></article>
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||||
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||||
<div class="autopermalink" data-description="Exercise Group 1.3.3.1–2"><a href="#exercises-Conditional-Probability-1" title="Copy heading and permalink for Exercise Group 1.3.3.1–2" aria-label="Copy heading and permalink for Exercise Group 1.3.3.1–2">🔗</a></div>
|
||||
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||||
<article class="exercise exercise-like" id="exercises-Conditional-Probability-2"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="introduction" id="exercises-Conditional-Probability-2-1"><div class="para" id="exercises-Conditional-Probability-2-1-1">A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div>
|
||||
<article class="task exercise-like" id="exercises-Conditional-Probability-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||||
<div class="para" id="exercises-Conditional-Probability-2-2-1-1">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
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||||
<div class="solutions"><details id="exercises-Conditional-Probability-2-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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||||
<div class="para" id="exercises-Conditional-Probability-2-2-2-1">TODO<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.3.3.3.a"><a href="#exercises-Conditional-Probability-2-2" title="Copy heading and permalink for Task 1.3.3.3.a" aria-label="Copy heading and permalink for Task 1.3.3.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Conditional-Probability-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||||
<div class="para" id="exercises-Conditional-Probability-2-3-1-1">For a patient who receives a negative test, what is the probability they do not have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exercises-Conditional-Probability-2-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-Conditional-Probability-2-3-2-1">TODO<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Conditional-Probability-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
|
||||
</ul></nav></div>
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Continuous-RVs"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Continuous Random Variables</span>
|
||||
</h2>
|
||||
<section class="subsection" id="subsec-pdfs"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Probability Density Functions</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-pdfs-2">We have to think about probability differently for continuous random variables. If <span class="process-math">\(X\)</span> is a continuous random variable taking values in an interval <span class="process-math">\([a, b]\text{,}\)</span> it doesn’t quite make sense to ask, as we have for discrete random variables, to find the probabilities of each individual outcome.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-pdfs-3">When we roll a fair 6-sided die, there are 6 equally likely outcomes, so they each get assigned the probability <span class="process-math">\(1/6\text{.}\)</span> If we try to select a random number in the interval <span class="process-math">\([0, 1]\)</span> in a similarly "fair" way (i.e., with each outcome equally likely), then what probability <span class="process-math">\(\alpha\)</span> should we assign to each outcome? If <span class="process-math">\(\alpha > 0\text{,}\)</span> then we have a problem: if you add up the probabilities of all the (infinitely many) outcomes, you get an infinite total, whereas the total probability in any sample space should be 1. If <span class="process-math">\(\alpha = 0\text{,}\)</span> then we also have a problem: if you add up the probabilities of all the outcomes, you get total of 0, whereas the total probability in any sample space should be 1.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-pdfs-4">We need a new way to model and calculate probabilities for continuous random variables.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-pdf"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-pdf-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a continuous random variable taking values in the interval <span class="process-math">\([a, b]\text{.}\)</span> The <dfn class="terminology">probability density function</dfn> (or <dfn class="terminology">pdf</dfn>) for <span class="process-math">\(X\)</span> is a function <span class="process-math">\(f(x)\)</span> defined on <span class="process-math">\([a, b]\)</span> satisfying the properties:</div>
|
||||
<ol class="decimal" id="def-pdf-1-1-8">
|
||||
<li id="def-pdf-1-1-8-1">
|
||||
<div class="para" id="def-pdf-1-1-8-1-1">
|
||||
<span class="process-math">\(f(x) \geq 0\)</span> for all <span class="process-math">\(x \in [a, b]\text{,}\)</span> and<div class="autopermalink" data-description="Paragraph"><a href="#def-pdf-1-1-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#def-pdf-1-1-8-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-pdf-1-1-8-2">
|
||||
<div class="para" id="def-pdf-1-1-8-2-1">
|
||||
<span class="process-math">\(\displaystyle{\int_a^b f(x)\ dx = 1}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-pdf-1-1-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#def-pdf-1-1-8-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-pdf-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.2.1"><a href="#def-pdf" title="Copy heading and permalink for Definition 2.2.1" aria-label="Copy heading and permalink for Definition 2.2.1">🔗</a></div></article><div class="para" id="subsec-pdfs-6">It’s important to keep in mind that the output values of a pdf <span class="process-math">\(f(x)\)</span> are <em class="emphasis">probability densities</em>, not probabilities. You can (somewhat casually) think of the units of <span class="process-math">\(f(x)\)</span> values as "probability per <units of the sample space>". You may not have discussed the units of the differential <span class="process-math">\(dx\)</span> very much in your Calculus 1 course, but <span class="process-math">\(dx\)</span> has the same units as <span class="process-math">\(x\)</span> and is being multiplied by the integrand when you take an integral. So, although <span class="process-math">\(f(x)\)</span> is not a probability, the result of computing <span class="process-math">\(\int_c^d f(x)\ dx\)</span> <em class="emphasis">is</em> a probability.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical" id="subsec-pdfs-7">
|
||||
<div class="para">A pdf’s purpose in life is to be integrated, which tells us probabilities for a continuous random variable:</div>
|
||||
<div class="displaymath process-math" id="subsec-pdfs-7-1">
|
||||
\begin{gather*}
|
||||
\Pr(c \leq X \leq d) = \int_c^d f(x)\ dx.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-continuous-uniform-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-continuous-uniform-distribution-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> take values in the interval <span class="process-math">\([a, b]\text{.}\)</span> We say that <span class="process-math">\(X\)</span> has the <dfn class="terminology">uniform distribution</dfn> if, for any subinterval <span class="process-math">\([c, d] \subset [a, b]\text{,}\)</span> the probability that <span class="process-math">\(X\)</span> lies within <span class="process-math">\([c, d]\)</span> is equal to the proportion of the total sample space taken by <span class="process-math">\([c, d]\text{.}\)</span> That is:</div>
|
||||
<div class="displaymath process-math" id="def-continuous-uniform-distribution-1-1-9">
|
||||
\begin{gather*}
|
||||
\Pr(c \leq X \leq d) = \frac{d - c}{b - a}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">In this case, <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-continuous-uniform-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.2.2"><a href="#def-continuous-uniform-distribution" title="Copy heading and permalink for Definition 2.2.2" aria-label="Copy heading and permalink for Definition 2.2.2">🔗</a></div></article><div class="para logical" id="subsec-pdfs-9">
|
||||
<div class="para">We’ll ignore for the moment how to find this pdf and instead we’ll quickly check that it behaves the way we want it to:</div>
|
||||
<div class="displaymath process-math" id="subsec-pdfs-9-1">
|
||||
\begin{align*}
|
||||
\Pr(c \leq X \leq d) \amp = \int_c^d \frac{1}{b-a}\ dx \\
|
||||
\amp = \frac{x}{b - a}\bigg|_c^d \\
|
||||
\amp = \frac{d - c}{b - a},
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">as we wanted.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-9" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsec-pdfs-10"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-pdfs-10-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a continuous random variable taking values in the interval <span class="process-math">\([0, 4]\)</span> with pdf <span class="process-math">\(f(x) = k \sqrt{x}\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> Because of the restriction that the total probability must be 1, we can determine the value of <span class="process-math">\(k\text{:}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="subsec-pdfs-10-1-1-6">
|
||||
\begin{align*}
|
||||
1 = \Pr(0\leq X \leq 4) \amp = \int_0^4 k\sqrt{x}\ dx \\
|
||||
\amp = \frac{kx^{3/2}}{3/2}\bigg|_0^4 \\
|
||||
\amp = \frac{2kx^{3/2}}{3}\bigg|_0^4 \\
|
||||
\amp = \left(\frac{2k(4^{3/2})}{3}\right) - 0 \\
|
||||
\amp = \frac{16k}{3}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Therefore <span class="process-math">\(k = \frac{3}{16}\text{.}\)</span> Now we can calculate probabilities, e.g.:</div>
|
||||
<div class="displaymath process-math" id="subsec-pdfs-10-1-1-8">
|
||||
\begin{align*}
|
||||
\Pr(1 \leq X \leq 2) \amp = \int_1^2 \frac{3\sqrt{x}}{16}\ dx \\
|
||||
\amp = \frac{3}{16}\left(\frac{2x^{3/2}}{3}\right)\bigg|_1^2 \\
|
||||
\amp = \frac{1}{8}\left(x^{3/2}\right)\bigg|_1^2 \\
|
||||
\amp = \frac{1}{8}\left( 2^{3/2} - 1 \right) \\
|
||||
\amp \approx 0.23.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-pdfs-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 2.2.3"><a href="#subsec-pdfs-10" title="Copy heading and permalink for Example 2.2.3" aria-label="Copy heading and permalink for Example 2.2.3">🔗</a></div></article><div class="autopermalink" data-description="Subsection 2.2.1: Probability Density Functions"><a href="#subsec-pdfs" title="Copy heading and permalink for Subsection 2.2.1: Probability Density Functions" aria-label="Copy heading and permalink for Subsection 2.2.1: Probability Density Functions">🔗</a></div></section><section class="subsection" id="subsec-cdf"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Cumulative Distribution Functions</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-cdf-2">When a random variable takes values in an interval <span class="process-math">\([a, b]\text{,}\)</span> it’s often useful to consider how much probability has been accumulated up to some specified value <span class="process-math">\(x \in [a, b]\text{.}\)</span> For each value of <span class="process-math">\(x\text{,}\)</span> we can calculate this accumulated probability. Since there’s an association between inputs (the <span class="process-math">\(x\)</span> values) and output (the accumulated probability), we can naturally build a function.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-cdf-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-cdf"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-cdf-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a continuous random variable taking values in an interval <span class="process-math">\([a, b]\text{.}\)</span> The <dfn class="terminology">cumulative distribution function</dfn> (or <dfn class="terminology">cdf</dfn>) is the function <span class="process-math">\(F(x)\)</span> defined on <span class="process-math">\([a, b]\)</span> such that</div>
|
||||
<div class="displaymath process-math" id="def-cdf-1-1-7">
|
||||
\begin{gather*}
|
||||
F(x) = \Pr(a \leq X \leq x).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-cdf-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.2.4"><a href="#def-cdf" title="Copy heading and permalink for Definition 2.2.4" aria-label="Copy heading and permalink for Definition 2.2.4">🔗</a></div></article><div class="para logical" id="subsec-cdf-4">
|
||||
<div class="para">How do we actually calculate these cumulative probabilities? Luckily, these are precisely the sort of probabilities we can use the pdf to find:</div>
|
||||
<div class="displaymath process-math" id="subsec-cdf-4-1">
|
||||
\begin{gather*}
|
||||
F(x) = \Pr(a \leq X \leq x) = \int_a^x f(t)\ dt.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">(Note that on the far left side of the equation, we used the letter <span class="process-math">\(x\)</span> to refer to some point within <span class="process-math">\([a, b]\)</span> up to which we’re trying to accumulate probability. So that value <span class="process-math">\(x\)</span> must also appear at the top of the integral sign. We can’t then use the same letter for the integrand and differential inside the integral; we can’t use the same letter to refer to two different variables at the same time. So, we have changed the letter inside the integral to <span class="process-math">\(t\text{.}\)</span>)</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-cdf-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical" id="subsec-cdf-5">
|
||||
<div class="para">This equation may look familiar from Calculus 1. If you review <a class="external" href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#First_part" target="_blank">the First Fundamental Theorem of Calculus</a>, you’ll see precisely the relationship between the cdf <span class="process-math">\(F(x)\)</span> and the pdf <span class="process-math">\(f(x)\text{:}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="subsec-cdf-5-4">
|
||||
\begin{gather*}
|
||||
F'(x) = f(x).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Let’s see how we can put this relationship to use.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-cdf-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="subsec-cdf-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="subsec-cdf-6-1-1">
|
||||
<div class="para">Suppose <span class="process-math">\(X\)</span> is uniformly distributed in the interval <span class="process-math">\([a, b]\text{.}\)</span> We previously said in <a href="sec-Continuous-RVs.html#def-continuous-uniform-distribution" class="xref" data-knowl="./knowl/xref/def-continuous-uniform-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.2.2">Definition 2.2.2</a> that, for any subinterval <span class="process-math">\([c, d] \subset [a, b]\text{,}\)</span> we should have</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-continuous-uniform-distribution.html" id="subsec-cdf-6-1-1-5">
|
||||
\begin{gather*}
|
||||
\Pr(c\leq X \leq d) = \frac{d - c}{b - a}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">In particular, for any <span class="process-math">\(x\in [a, b]\text{,}\)</span> we would have:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-continuous-uniform-distribution.html" id="subsec-cdf-6-1-1-7">
|
||||
\begin{gather*}
|
||||
F(x) = \Pr(a \leq X \leq x) = \frac{x - a}{b - a}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Therefore, taking the derivative:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-continuous-uniform-distribution.html" id="subsec-cdf-6-1-1-8">
|
||||
\begin{gather*}
|
||||
f(x) = F'(x) = \frac{1}{b - a}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-cdf-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 2.2.5"><a href="#subsec-cdf-6" title="Copy heading and permalink for Example 2.2.5" aria-label="Copy heading and permalink for Example 2.2.5">🔗</a></div></article><div class="para" id="subsec-cdf-7">So, we can see that the idea we wanted to capture in probability led to a formula that gave accumulated probability, which then led to the pdf. Both the cdf and the pdf contain essentially all of the information we need to answer probability questions about <span class="process-math">\(X\text{.}\)</span> However, keep in mind that these functions output different mathematical objects. <span class="process-math">\(F(x)\)</span> outputs probabilities already; <span class="process-math">\(f(x)\)</span> must be integrated over a subinterval to find probabilities.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-cdf-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection 2.2.2: Cumulative Distribution Functions"><a href="#subsec-cdf" title="Copy heading and permalink for Subsection 2.2.2: Cumulative Distribution Functions" aria-label="Copy heading and permalink for Subsection 2.2.2: Cumulative Distribution Functions">🔗</a></div></section><section class="subsection" id="subsec-exponential-distribution"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">The Exponential Distribution</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="def-exponential-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-exponential-distribution-1-1">Suppose we observe a Poisson process in which an event occurs with rate parameter <span class="process-math">\(\lambda\text{.}\)</span> Let <span class="process-math">\(T\)</span> be the time until the event is first observed. Then <span class="process-math">\(T\)</span> has the <dfn class="terminology">exponential distribution</dfn> with parameter <span class="process-math">\(\lambda\text{.}\)</span> We’ll write <span class="process-math">\(T \sim \Exp(\lambda)\)</span> to denote this.<div class="autopermalink" data-description="Paragraph"><a href="#def-exponential-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.2.6"><a href="#def-exponential-distribution" title="Copy heading and permalink for Definition 2.2.6" aria-label="Copy heading and permalink for Definition 2.2.6">🔗</a></div></article><article class="fact theorem-like" id="fact-exponential-pdf"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exponential Density Function.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="fact-exponential-pdf-2-1">
|
||||
<div class="displaymath process-math" id="fact-exponential-pdf-2-1-1">
|
||||
\begin{gather*}
|
||||
f(t) = \lambda e^{-\lambda t}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-exponential-pdf-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 2.2.7: Exponential Density Function"><a href="#fact-exponential-pdf" title="Copy heading and permalink for Fact 2.2.7: Exponential Density Function" aria-label="Copy heading and permalink for Fact 2.2.7: Exponential Density Function">🔗</a></div></article><div class="para logical" id="subsec-exponential-distribution-4">
|
||||
<div class="para">The binomial, geometric, Poisson, and exponential distributions are conceptually grouped together. Consider the following two dichotomies:</div>
|
||||
<ol class="decimal" id="subsec-exponential-distribution-4-1">
|
||||
<li id="subsec-exponential-distribution-4-1-1">
|
||||
<div class="para" id="subsec-exponential-distribution-4-1-1-1">Is time being measured discretely or continuously?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#subsec-exponential-distribution-4-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="subsec-exponential-distribution-4-1-2">
|
||||
<div class="para" id="subsec-exponential-distribution-4-1-2-1">Does the random variable count occurrences of an event, or does it measure time until an event?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 2"><a href="#subsec-exponential-distribution-4-1-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="para">There are four combinations of answers to these two questions.</div>
|
||||
<dl class="description-list">
|
||||
<dt id="subsec-exponential-distribution-4-2-1">Binomial<div class="autopermalink" data-description="Item: Binomial"><a href="#subsec-exponential-distribution-4-2-1" title="Copy heading and permalink for Item: Binomial" aria-label="Copy heading and permalink for Item: Binomial">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="subsec-exponential-distribution-4-2-1-2">"Time" is discrete; the random variables counts occurrences.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-2-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="subsec-exponential-distribution-4-2-2">Geometric<div class="autopermalink" data-description="Item: Geometric"><a href="#subsec-exponential-distribution-4-2-2" title="Copy heading and permalink for Item: Geometric" aria-label="Copy heading and permalink for Item: Geometric">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="subsec-exponential-distribution-4-2-2-2">"Time" is discrete; the random variable measures "time" until an event.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-2-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="subsec-exponential-distribution-4-2-3">Poisson<div class="autopermalink" data-description="Item: Poisson"><a href="#subsec-exponential-distribution-4-2-3" title="Copy heading and permalink for Item: Poisson" aria-label="Copy heading and permalink for Item: Poisson">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="subsec-exponential-distribution-4-2-3-2">Time is continuous; the random variable counts occurrences.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-2-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="subsec-exponential-distribution-4-2-4">Exponential<div class="autopermalink" data-description="Item: Exponential"><a href="#subsec-exponential-distribution-4-2-4" title="Copy heading and permalink for Item: Exponential" aria-label="Copy heading and permalink for Item: Exponential">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="subsec-exponential-distribution-4-2-4-2">Time is continuous; the random variable measures time until an event.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4-2-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
</dl>
|
||||
<div class="para">When we talk about discrete "time", we don’t always mean something measured in actual units of time. We might; for example, an experiment may involve a bacterial colony growing in a petri dish over time, but we only check in and measure something about the bacterial colony once each day. In this case, since our measurements of time must occur in whole numbers of days, time is being measured in a discrete way (that is, not involving an interval of real numbers).</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-exponential-distribution-5">Consider, instead, an experiment in which we flip a coin repeatedly until we first see heads appear. The number of flips doesn’t have units of time, but counting flips is analogous to measuring time until an event occurs. The flip count is a discrete stand-in for something analogous to time. We can consider this experiment to be a discrete analog of a Poisson process.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-exponential-distribution-6">It’s also worth including a warning here about the Poisson distribution, specifically. The dichotomy mentioned above about whether time is being measured discretely or continuously is <em class="emphasis">not</em> the same as the consideration of whether the random variable itself is discrete or continuous. In a Poisson process, time is being measured continuously. But the Poisson random variable counts the number of occurrences of the event, and this number must be a whole number. So a Poisson random variable is a discrete random variable.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-exponential-distribution-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection 2.2.3: The Exponential Distribution"><a href="#subsec-exponential-distribution" title="Copy heading and permalink for Subsection 2.2.3: The Exponential Distribution" aria-label="Copy heading and permalink for Subsection 2.2.3: The Exponential Distribution">🔗</a></div></section><section class="subsection" id="subsec-normal-distribution"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Normal Distribution</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="def-normal-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.8</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-normal-distribution-1-1">
|
||||
<div class="para">Consider the parametrized family of functions:</div>
|
||||
<div class="displaymath process-math" id="def-normal-distribution-1-1-1">
|
||||
\begin{gather*}
|
||||
f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">The function <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is called a <dfn class="terminology">normal density function</dfn> with parameters <span class="process-math">\(\mu\)</span> and <span class="process-math">\(\sigma^2\text{.}\)</span> A continuous random variable <span class="process-math">\(X\)</span> with pdf <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is said to have a <dfn class="terminology">normal distribution</dfn>. In the specific case that <span class="process-math">\(\mu = 0\)</span> and <span class="process-math">\(\sigma^2 = 1\text{,}\)</span> we call the resulting distribution the <dfn class="terminology">standard normal distribution</dfn>.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-normal-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.2.8"><a href="#def-normal-distribution" title="Copy heading and permalink for Definition 2.2.8" aria-label="Copy heading and permalink for Definition 2.2.8">🔗</a></div></article><div class="para" id="subsec-normal-distribution-3">It’s an unfortunate reality that the normal density functions can’t easily be integrated. (In fact, it is provably impossible to write nice formulas for their antiderivatives). So we won’t worry much about probability computations like <span class="process-math">\(\Pr(c \leq X \leq d)\)</span> for normal distributions. The normal distributions will play a <em class="emphasis">central</em> role in later chapters (after which you can return here and appreciate this pun).<div class="autopermalink" data-description="Paragraph"><a href="#subsec-normal-distribution-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection 2.2.4: Normal Distribution"><a href="#subsec-normal-distribution" title="Copy heading and permalink for Subsection 2.2.4: Normal Distribution" aria-label="Copy heading and permalink for Subsection 2.2.4: Normal Distribution">🔗</a></div></section><section class="exercises" id="exercises-Continuous-RVs"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.5</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Continuous-RVs-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Continuous-RVs-1-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 2.2.5.1"><a href="#exercises-Continuous-RVs-1" title="Copy heading and permalink for Exercise 2.2.5.1" aria-label="Copy heading and permalink for Exercise 2.2.5.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Continuous-RVs-2-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find the c.d.f. <span class="process-math">\(F(x)\text{.}\)</span> Use your c.d.f. to find <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 2.2.5.2"><a href="#exercises-Continuous-RVs-2" title="Copy heading and permalink for Exercise 2.2.5.2" aria-label="Copy heading and permalink for Exercise 2.2.5.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Continuous-RVs-3-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([2, 3]\)</span> has c.d.f. <span class="process-math">\(F(x) = \frac{x^3}{3} - x^2 + 4\text{.}\)</span> Find the p.d.f. <span class="process-math">\(f(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 2.2.5.3"><a href="#exercises-Continuous-RVs-3" title="Copy heading and permalink for Exercise 2.2.5.3" aria-label="Copy heading and permalink for Exercise 2.2.5.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Continuous-RVs-4-1-1">A radioactive material emits particles at an unknown probabilistic rate <span class="process-math">\(\lambda\)</span> particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function <span class="process-math">\(\mathcal{L}(\lambda)\)</span> based on this data. What is the maximum likelihood estimation for <span class="process-math">\(\lambda\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 2.2.5.4"><a href="#exercises-Continuous-RVs-4" title="Copy heading and permalink for Exercise 2.2.5.4" aria-label="Copy heading and permalink for Exercise 2.2.5.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Continuous-RVs-5-1-1">Suppose a parameter <span class="process-math">\(\theta\)</span> takes values in <span class="process-math">\([0, 1]\)</span> with likelihood function <span class="process-math">\(\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 2.2.5.5"><a href="#exercises-Continuous-RVs-5" title="Copy heading and permalink for Exercise 2.2.5.5" aria-label="Copy heading and permalink for Exercise 2.2.5.5">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.2.5"><a href="#exercises-Continuous-RVs" title="Copy heading and permalink for Exercises 2.2.5" aria-label="Copy heading and permalink for Exercises 2.2.5">🔗</a></div></section><div class="autopermalink" data-description="Section 2.2: Continuous Random Variables"><a href="#sec-Continuous-RVs" title="Copy heading and permalink for Section 2.2: Continuous Random Variables" aria-label="Copy heading and permalink for Section 2.2: Continuous Random Variables">🔗</a></div></section></div>
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
|
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
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<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
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</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
|
||||
<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
|
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<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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||||
<ul class="structural toc-item-list contains-active">
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||||
<li class="toc-item toc-section contains-active">
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||||
<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Discrete-RVs"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Discrete Random Variables</span>
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</h2>
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<section class="subsection" id="subsec-Discrete-RVs"><h3 class="heading hide-type">
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<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Discrete Random Variables</span>
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</h3>
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<article class="definition definition-like" id="def-RV"><h4 class="heading">
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||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
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</h4>
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||||
<div class="para" id="def-RV-1-1">Let <span class="process-math">\(\Omega\)</span> be a sample space. A <dfn class="terminology">random variable</dfn> is a function <span class="process-math">\(X \colon \Omega \to \R\)</span> which assigns a number to each outcome.<div class="autopermalink" data-description="Paragraph"><a href="#def-RV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.1"><a href="#def-RV" title="Copy heading and permalink for Definition 2.1.1" aria-label="Copy heading and permalink for Definition 2.1.1">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-RVs-3"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Discrete-RVs-3-1-1">Suppose an experiment consists of flipping a fair coin 4 times. Let <span class="process-math">\(X\)</span> be the number of heads. The actual outcomes in the experiment are heads/tails sequences of length 4, such as <span class="process-math">\(HTHT\)</span> and <span class="process-math">\(HHHT\text{.}\)</span> The random variable <span class="process-math">\(X\)</span> assigns a numerical measurement to the outcomes, such as <span class="process-math">\(X(HTHT) = 2\)</span> and <span class="process-math">\(X(HHHT) = 3\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsec-Discrete-RVs-3-1-2">For any real number value <span class="process-math">\(a\text{,}\)</span> we can consider the event consisting of outcomes such that <span class="process-math">\(X = a\text{.}\)</span> For example, the flip sequences <span class="process-math">\(HHHT, HHTH, HTHH\text{,}\)</span> and <span class="process-math">\(THHH\)</span> all have 3 heads, so <span class="process-math">\(\Pr(X = 3) = \frac{4}{16} = \frac{1}{4}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 2.1.2"><a href="#subsec-Discrete-RVs-3" title="Copy heading and permalink for Example 2.1.2" aria-label="Copy heading and permalink for Example 2.1.2">🔗</a></div></article><article class="example example-like" id="example-indicator-distribution"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Indicator Random Variable.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="example-indicator-distribution-2-1">
|
||||
<div class="para">Let <span class="process-math">\(A\)</span> be an event in the sample space <span class="process-math">\(\Omega\text{.}\)</span> An <dfn class="terminology">indicator random variable</dfn> for <span class="process-math">\(A\)</span> is the random variable <span class="process-math">\(X\)</span> such that</div>
|
||||
<div class="displaymath process-math" id="example-indicator-distribution-2-1-6">
|
||||
\begin{align*}
|
||||
X(\omega) = \begin{cases} 1 \amp \omega \in A \\ 0 \amp \omega \notin A \end{cases}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-distribution-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 2.1.3: Indicator Random Variable"><a href="#example-indicator-distribution" title="Copy heading and permalink for Example 2.1.3: Indicator Random Variable" aria-label="Copy heading and permalink for Example 2.1.3: Indicator Random Variable">🔗</a></div></article><div class="para" id="subsec-Discrete-RVs-5">Indicator random variables are also called <dfn class="terminology">Bernoulli random variables</dfn>, although we’ll prefer the former term since it more clearly states the purpose of these random variables: to indicate whether or not a particular event has occurred. We’ll use the language "<span class="process-math">\(X\)</span> indicates <span class="process-math">\(A\)</span>" to mean that <span class="process-math">\(X\)</span> is an indicator random variable for the event <span class="process-math">\(A\text{.}\)</span> In this case, <span class="process-math">\(\Pr(X = 1) = \Pr(A)\text{,}\)</span> and <span class="process-math">\(\Pr(X = 0) = 1 - \Pr(A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-discrete-and-continuous"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-discrete-and-continuous-1-1">Consider the set <span class="process-math">\(S\)</span> of all real numbers which are actually output by a random variable <span class="process-math">\(X\text{.}\)</span> If <span class="process-math">\(S\)</span> does not contain any interval of values, then the random variable <span class="process-math">\(X\)</span> is called <dfn class="terminology">discrete</dfn>. Otherwise, it’s called <dfn class="terminology">continuous</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-and-continuous-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.4"><a href="#def-discrete-and-continuous" title="Copy heading and permalink for Definition 2.1.4" aria-label="Copy heading and permalink for Definition 2.1.4">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-RVs-7"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-Discrete-RVs-7-1-1">Any random variable <span class="process-math">\(X\)</span> defined on a finite sample space <span class="process-math">\(\Omega\)</span> is discrete—the set of outputs of <span class="process-math">\(X\)</span> cannot contain an interval if it only has finitely many values.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsec-Discrete-RVs-7-1-2">Suppose an experiment consists of growing a plant in a new fertilizer, and a random variable <span class="process-math">\(Y\)</span> measures the height of the plant after a set growing period. Now <span class="process-math">\(Y\)</span> could conceivably take on an interval’s worth of values (such as, e.g., any real number between 10 inches and 20 inches), so this random variable would be continuous.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="subsec-Discrete-RVs-7-1-3">The distinction between discrete and continuous random variables is not simply the distinction of whether the sample space is finite or infinite. Consider the experiment in which we flip a coin repeatedly until we first see a coin come up heads. Let <span class="process-math">\(Z\)</span> be the number of times the coin is flipped. Then there are infinitely many possible values of <span class="process-math">\(Z\)</span> (1, 2, 3, and so on), but there’s no interval of real numbers which are all possible outputs of <span class="process-math">\(Z\text{.}\)</span> The possible outputs of <span class="process-math">\(Z\)</span> are discrete (in the non-technical, English sense of the word: separated).<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-7-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 2.1.5"><a href="#subsec-Discrete-RVs-7" title="Copy heading and permalink for Example 2.1.5" aria-label="Copy heading and permalink for Example 2.1.5">🔗</a></div></article><div class="para" id="subsec-Discrete-RVs-8">There are several particular types of distributions that appear repeatedly, since they model common behaviors. Indicator random variables, for example, will appear any time a random process generates a yes/no or success/failure type of answer to a question. Below are some other common distributions.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-RVs-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection 2.1.1: Discrete Random Variables"><a href="#subsec-Discrete-RVs" title="Copy heading and permalink for Subsection 2.1.1: Discrete Random Variables" aria-label="Copy heading and permalink for Subsection 2.1.1: Discrete Random Variables">🔗</a></div></section><section class="subsection" id="subsec-binomial-distribution"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">The Binomial Distribution</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="def-binomial-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-binomial-distribution-1-1">Suppose an event occurs with probability <span class="process-math">\(p\text{.}\)</span> If we perform <span class="process-math">\(n\)</span> independent trials, let <span class="process-math">\(S\)</span> be the random variable which counts the number of trials in which the event occurred. Then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> We’ll write <span class="process-math">\(S \sim \Bin(n, p)\)</span> to denote this. For each value <span class="process-math">\(0 \leq k \leq n\text{,}\)</span> we’ll write <span class="process-math">\(b(k)\)</span> for <span class="process-math">\(\Pr(S = k)\text{.}\)</span> (If we want to keep track of the parameter values, we may write <span class="process-math">\(b(k; n, p)\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#def-binomial-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.6"><a href="#def-binomial-distribution" title="Copy heading and permalink for Definition 2.1.6" aria-label="Copy heading and permalink for Definition 2.1.6">🔗</a></div></article><article class="example example-like" id="subsec-binomial-distribution-3"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="subsec-binomial-distribution-3-1-1">Suppose we have a coin which comes up heads with probability <span class="process-math">\(p = 0.4\text{.}\)</span> If we flip the coin three times and let <span class="process-math">\(S\)</span> be the number of heads, then <span class="process-math">\(S\)</span> is binomially distributed with parameters <span class="process-math">\(n = 3\)</span> and <span class="process-math">\(p = 0.4\text{.}\)</span> Then <span class="process-math">\(S\)</span> has the distribution below.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-binomial-distribution-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="table-binomial-example"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Bin(3, 0.4)\)</span><div class="autopermalink" data-description="Table 2.1.8: \Bin(3, 0.4)"><a href="#table-binomial-example" title="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)" aria-label="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">0</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.216</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.432</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.288</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.064</td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Example 2.1.7"><a href="#subsec-binomial-distribution-3" title="Copy heading and permalink for Example 2.1.7" aria-label="Copy heading and permalink for Example 2.1.7">🔗</a></div></article><div class="para logical" id="subsec-binomial-distribution-4">
|
||||
<div class="para">To understand how these probabilities are calculated, consider a particular heads/tails sequence, such as <span class="process-math">\(HHHT\text{.}\)</span> If the probability of heads is 0.4, then the probability of tails is 0.6. Since the results of different coin flips are independent from each other, the probability of seeing precisely the sequence <span class="process-math">\(HHT\)</span> would be the product <span class="process-math">\((0.4)(0.4)(0.6) = 0.096\text{.}\)</span> The probability of seeing precisely <span class="process-math">\(HTH\)</span> would be <span class="process-math">\((0.4)(0.6)(0.4) = 0.096\text{,}\)</span> the same product of terms in a different order. So, if we knew exactly the number of flip sequences of length 3 with exactly 2 heads, we could calculate the probability of seeing 2 heads as:</div>
|
||||
<div class="displaymath process-math" id="subsec-binomial-distribution-4-6">
|
||||
\begin{gather*}
|
||||
\Pr(S = 2) = (\text{number of sequences})(\text{probability of each sequence})
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">The probability for each sequence is straightforward to generalize: if there are <span class="process-math">\(n\)</span> flips and <span class="process-math">\(k\)</span> of them are heads, then <span class="process-math">\(n - k\)</span> of them are tails. So the product we should calculate will have <span class="process-math">\(k\)</span> copies of the parameter <span class="process-math">\(p\)</span> and <span class="process-math">\(n - k\)</span> copies of the complementary probability <span class="process-math">\(1 - p\text{:}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="subsec-binomial-distribution-4-14">
|
||||
\begin{gather*}
|
||||
\text{probability of each sequence } = p^k(1-p)^k
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-binomial-distribution-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-binomial-coefficient"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.9</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-binomial-coefficient-1-1">We’ll write the symbol <span class="process-math">\({n \choose k}\)</span> to mean the number of ways to choose <span class="process-math">\(k\)</span> things out of <span class="process-math">\(n\)</span> things. The number <span class="process-math">\({n \choose k}\)</span> is read "<span class="process-math">\(n\)</span> choose <span class="process-math">\(k\)</span>" and called a <dfn class="terminology">binomial coefficient</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-binomial-coefficient-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.9"><a href="#def-binomial-coefficient" title="Copy heading and permalink for Definition 2.1.9" aria-label="Copy heading and permalink for Definition 2.1.9">🔗</a></div></article><div class="para" id="subsec-binomial-distribution-6">The term "binomial coefficient" is used here because these numbers form coefficients of a polynomial in an important theorem about binomials called the <a class="external" href="https://en.wikipedia.org/wiki/Binomial_theorem" target="_blank">Binomial Theorem</a>. The binomial coefficients will precisely tell us the number of flip sequences of length <span class="process-math">\(n\)</span> with precisely <span class="process-math">\(k\)</span> heads (to build such a sequence, we must choose <span class="process-math">\(k\)</span> out of <span class="process-math">\(n\)</span> flips in the sequence to be heads; the rest must be tails). So, now we need a way to find the numbers <span class="process-math">\({n \choose k}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-binomial-distribution-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-factorial"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.10</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-factorial-1-1">
|
||||
<div class="para">Let <span class="process-math">\(n\)</span> be a positive integer. Then the <dfn class="terminology">factorial</dfn> of <span class="process-math">\(n\text{,}\)</span> written <span class="process-math">\(n!\text{,}\)</span> is the product of the positive integers up to <span class="process-math">\(n\text{,}\)</span> i.e.:</div>
|
||||
<div class="displaymath process-math" id="def-factorial-1-1-6">
|
||||
\begin{gather*}
|
||||
n! = 1 \times 2 \times 3 \times \dotsb \times (n-1) \times n.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-factorial-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.10"><a href="#def-factorial" title="Copy heading and permalink for Definition 2.1.10" aria-label="Copy heading and permalink for Definition 2.1.10">🔗</a></div></article><article class="fact theorem-like" id="fact-binomial-coefficient-formula"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Binomial Coefficient Formula.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="fact-binomial-coefficient-formula-2-1">
|
||||
<div class="para">For any <span class="process-math">\(0 \leq k \leq n\text{,}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="fact-binomial-coefficient-formula-2-1-2">
|
||||
\begin{gather*}
|
||||
{n \choose k} = \frac{n!}{k!(n-k)!}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-binomial-coefficient-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 2.1.11: Binomial Coefficient Formula"><a href="#fact-binomial-coefficient-formula" title="Copy heading and permalink for Fact 2.1.11: Binomial Coefficient Formula" aria-label="Copy heading and permalink for Fact 2.1.11: Binomial Coefficient Formula">🔗</a></div></article><div class="para logical" id="subsec-binomial-distribution-9">
|
||||
<div class="para">We won’t provide a justification of this formula here, but we’ll put it to use to calculate probabilities in the binomial distribution. For example, the number of flip sequences of length 3 with exactly 2 heads is:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/table-binomial-example.html" id="subsec-binomial-distribution-9-1">
|
||||
\begin{gather*}
|
||||
{3 \choose 2} = \frac{3!}{2!1!} = \frac{6}{2} = 3.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Since we previously said the probability of such a sequence is 0.096, we can now find the probability of seeing exactly 2 heads in 3 flips of the coin:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/table-binomial-example.html" id="subsec-binomial-distribution-9-2">
|
||||
\begin{align*}
|
||||
\Pr(S = 2) \amp = (\text{number of sequences})(\text{probability of each sequence}) \\
|
||||
\amp = (3)(0.096) \\
|
||||
\amp = 0.288,
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">which matches the value in <a href="sec-Discrete-RVs.html#table-binomial-example" class="xref" data-knowl="./knowl/xref/table-binomial-example.html" data-reveal-label="Reveal" data-close-label="Close" title="Table 2.1.8: \Bin(3, 0.4)">Table 2.1.8</a>. More generally:</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-binomial-distribution-9" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="fact theorem-like" id="fact-binomial-probability-formula"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.12</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Binomial Probability Formula.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="fact-binomial-probability-formula-2-1">
|
||||
<div class="displaymath process-math" id="fact-binomial-probability-formula-2-1-1">
|
||||
\begin{gather*}
|
||||
b(k; n, p) = {n \choose k} p^k (1-p)^{n-k}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-binomial-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 2.1.12: Binomial Probability Formula"><a href="#fact-binomial-probability-formula" title="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula">🔗</a></div></article><div class="autopermalink" data-description="Subsection 2.1.2: The Binomial Distribution"><a href="#subsec-binomial-distribution" title="Copy heading and permalink for Subsection 2.1.2: The Binomial Distribution" aria-label="Copy heading and permalink for Subsection 2.1.2: The Binomial Distribution">🔗</a></div></section><section class="subsection" id="subsec-geometric-distribution"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Geometric Distribution</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="def-geometric-distribution"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.13</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-geometric-distribution-1-1">Suppose an event occurs with probability <span class="process-math">\(p\text{.}\)</span> If we perform independent trials repeatedly, let <span class="process-math">\(N\)</span> be the random variable which counts the number of trials performed until we see the event occur for the first time. (<span class="process-math">\(N\)</span> counts the final trial in which the event occurs.) Then <span class="process-math">\(N\)</span> has the <dfn class="terminology">geometric distribution</dfn> with parameter <span class="process-math">\(p\text{.}\)</span> We’ll write <span class="process-math">\(N \sim \Geom(p)\)</span> to denote this. For each value <span class="process-math">\(k \geq 1\text{,}\)</span> we’ll write <span class="process-math">\(g(k)\)</span> for <span class="process-math">\(\Pr(N = k)\text{.}\)</span> (If we want to keep track of the parameter value, we may write <span class="process-math">\(g(k; p)\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#def-geometric-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 2.1.13"><a href="#def-geometric-distribution" title="Copy heading and permalink for Definition 2.1.13" aria-label="Copy heading and permalink for Definition 2.1.13">🔗</a></div></article><div class="para" id="subsec-geometric-distribution-3">Suppose we flip a coin repeatedly until we first see heads. The number <span class="process-math">\(N\)</span> of flips is geometrically distributed with parameter <span class="process-math">\(p\text{,}\)</span> which gives the probability of a flip coming up heads. Then the probability of tails is <span class="process-math">\(1 - p\text{.}\)</span> For any particular <span class="process-math">\(k \geq 1\text{,}\)</span> in order to have <span class="process-math">\(N = k\text{,}\)</span> we must see exactly the flip sequence <span class="process-math">\(TT\dotsm TH\text{,}\)</span> where the number of tails is exactly <span class="process-math">\(k - 1\text{.}\)</span> The probability of seeing this particular sequence is then:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-geometric-distribution-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="fact theorem-like" id="fact-geometric-probability-formula"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.14</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Geometric Probability Formula.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="fact-geometric-probability-formula-2-1">
|
||||
<div class="displaymath process-math" id="fact-geometric-probability-formula-2-1-1">
|
||||
\begin{gather*}
|
||||
g(k; p) = (1-p)^{k-1} p
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-geometric-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Fact 2.1.14: Geometric Probability Formula"><a href="#fact-geometric-probability-formula" title="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula">🔗</a></div></article><div class="autopermalink" data-description="Subsection 2.1.3: Geometric Distribution"><a href="#subsec-geometric-distribution" title="Copy heading and permalink for Subsection 2.1.3: Geometric Distribution" aria-label="Copy heading and permalink for Subsection 2.1.3: Geometric Distribution">🔗</a></div></section><section class="subsection" id="subsec-poisson-distribution"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Poisson Distribution</span>
|
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</h3>
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<article class="definition definition-like" id="def-poisson-distribution"><h4 class="heading">
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||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.15</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="def-poisson-distribution-1-1">A <dfn class="terminology">Poisson process</dfn> is one in which some event occurs randomly at a constant probabilistic rate <span class="process-math">\(\lambda\)</span> over time. Suppose we observe a Poisson process for a fixed amount of time <span class="process-math">\(t\text{,}\)</span> and let <span class="process-math">\(N\)</span> count the number of occurrences of the event. Then <span class="process-math">\(N\)</span> has the <dfn class="terminology">Poisson distribution</dfn> with parameters <span class="process-math">\(\lambda\)</span> and <span class="process-math">\(t\text{.}\)</span> We’ll write <span class="process-math">\(N \sim \Poiss(\lambda, t)\)</span> to denote this. For each value <span class="process-math">\(k \geq 0\text{,}\)</span> we’ll write <span class="process-math">\(p(k)\)</span> for <span class="process-math">\(\Pr(N = k)\text{.}\)</span> (If we want to keep track of the parameter values, we may write <span class="process-math">\(p(k; \lambda, t)\text{.}\)</span> See <a href="sec-Discrete-RVs.html#example-poisson-rate" class="xref" data-knowl="./knowl/xref/example-poisson-rate.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 2.1.17">Example 2.1.17</a> for an important detail about units.)<div class="autopermalink" data-description="Paragraph"><a href="#def-poisson-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 2.1.15"><a href="#def-poisson-distribution" title="Copy heading and permalink for Definition 2.1.15" aria-label="Copy heading and permalink for Definition 2.1.15">🔗</a></div></article><article class="fact theorem-like" id="fact-poisson-probability-formula"><h4 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.16</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Poisson Probability Formula.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="fact-poisson-probability-formula-2-1">
|
||||
<div class="displaymath process-math" id="fact-poisson-probability-formula-2-1-1">
|
||||
\begin{gather*}
|
||||
p(k; \lambda, t) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-poisson-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Fact 2.1.16: Poisson Probability Formula"><a href="#fact-poisson-probability-formula" title="Copy heading and permalink for Fact 2.1.16: Poisson Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.16: Poisson Probability Formula">🔗</a></div></article><div class="para" id="subsec-poisson-distribution-4">The term <span class="process-math">\(\lambda t\)</span> is potentially misleading here depending on the units given for each value. The idea is that the rate information might be given for a different period of time than the observation period, so the rate should be scaled before calculating a probability. But the rate and observation time must be using the same time unit before multiplying. See the following example.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-poisson-distribution-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="example-poisson-rate"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="example-poisson-rate-1-1">
|
||||
<div class="para">Suppose we observe traffic along a particular stretch of highway which typically has 200 cars pass per hour. Let <span class="process-math">\(N\)</span> be the number of cars seen in a 2-hour observation period. Since the parameter <span class="process-math">\(\lambda = 200\)</span> cars per hour is given for a 1-hour period, we should scale it to find the rate for a 2-hour period: <span class="process-math">\(\lambda t = (200)(2) = 400\)</span> cars per 2-hours. Then, for example:</div>
|
||||
<div class="displaymath process-math" id="example-poisson-rate-1-1-4">
|
||||
\begin{gather*}
|
||||
\Pr(N = 375) = p(375; 200, 2) = \frac{400^{375}}{375!} e^{-400} \approx 0.01.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="example-poisson-rate-1-2">
|
||||
<div class="para">Now consider the random variable <span class="process-math">\(M\)</span> counting the number of cars seen in a 30-minute observation period. We should not calculate <span class="process-math">\(\lambda t = (200)(30) = 6000\text{,}\)</span> since <span class="process-math">\(\lambda\)</span> measures time in hours and <span class="process-math">\(t\)</span> measures time in minutes. We should first convert <span class="process-math">\(t = 0.5\)</span> hours, then we can find our appropriately scaled rate information: <span class="process-math">\((200)(0.5) = 100\)</span> cars per half-hour. Then, for example:</div>
|
||||
<div class="displaymath process-math" id="example-poisson-rate-1-2-7">
|
||||
\begin{gather*}
|
||||
\Pr(M = 110) = p(110; 200, 0.5) = \frac{100^{110}}{110!}e^{-100} \approx 0.02.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Note that when we write the notation <span class="process-math">\(p(k; \lambda, t)\text{,}\)</span> we assume that <span class="process-math">\(\lambda\)</span> and <span class="process-math">\(t\)</span> are expressed with the same time units already.</div>
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||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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||||
<div class="autopermalink" data-description="Example 2.1.17"><a href="#example-poisson-rate" title="Copy heading and permalink for Example 2.1.17" aria-label="Copy heading and permalink for Example 2.1.17">🔗</a></div></article><div class="autopermalink" data-description="Subsection 2.1.4: Poisson Distribution"><a href="#subsec-poisson-distribution" title="Copy heading and permalink for Subsection 2.1.4: Poisson Distribution" aria-label="Copy heading and permalink for Subsection 2.1.4: Poisson Distribution">🔗</a></div></section><section class="exercises" id="exercises-Discrete-RVs"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.5</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Discrete-RVs-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-1-1-1">We flip a coin with unknown bias 100 times and observe 43 heads. What is the maximum likelihood estimation for the probability of the coin coming up heads?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.1"><a href="#exercises-Discrete-RVs-1" title="Copy heading and permalink for Exercise 2.1.5.1" aria-label="Copy heading and permalink for Exercise 2.1.5.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-2-1-1">An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability <span class="process-math">\(p = 0.3\text{,}\)</span> find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.2"><a href="#exercises-Discrete-RVs-2" title="Copy heading and permalink for Exercise 2.1.5.2" aria-label="Copy heading and permalink for Exercise 2.1.5.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-3-1-1">An experiment consists of flipping a coin repeatedly until we first see heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.3"><a href="#exercises-Discrete-RVs-3" title="Copy heading and permalink for Exercise 2.1.5.3" aria-label="Copy heading and permalink for Exercise 2.1.5.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-4-1-1">If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.4"><a href="#exercises-Discrete-RVs-4" title="Copy heading and permalink for Exercise 2.1.5.4" aria-label="Copy heading and permalink for Exercise 2.1.5.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-5-1-1">Suppose the coin has an unknown probability of coming up heads. We perform the experiment in five independent trials, during which it takes 4, 5, 4, 3, and 6 flips to see our first heads in each trial. What is the maximum likelihood estimation for the probability of the coin coming up heads on a flip?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.5"><a href="#exercises-Discrete-RVs-5" title="Copy heading and permalink for Exercise 2.1.5.5" aria-label="Copy heading and permalink for Exercise 2.1.5.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Discrete-RVs-6-1-1">A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 2.1.5.6"><a href="#exercises-Discrete-RVs-6" title="Copy heading and permalink for Exercise 2.1.5.6" aria-label="Copy heading and permalink for Exercise 2.1.5.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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||||
<div class="para" id="exercises-Discrete-RVs-7-1-1">A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2 focused" data-preexpanded-levels="0" data-max-levels="2"><ul class="structural toc-item-list contains-active">
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<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
|
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<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
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|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
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|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
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|
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|
||||
</ul>
|
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|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
|
||||
<ul class="structural toc-item-list contains-active">
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
|
||||
<ul class="structural toc-item-list active">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
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|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
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|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
||||
</ul>
|
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</li>
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<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
|
||||
</ul>
|
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</li>
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
|
||||
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<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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</ul></nav></div>
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||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Expected-Value"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Expected Value</span>
|
||||
</h2>
|
||||
<section class="subsection" id="subsec-discrete-EV"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Discrete Expected Value</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-discrete-EV-2">A full probability distribution (for a discrete random variable) or a density function (for a continuous random variable) carry all of the probability information. Often, we seek <dfn class="terminology">statistics</dfn>, i.e., numbers that summarize a distribution in some way.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-discrete-expected-value"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-discrete-expected-value-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> (also called <dfn class="terminology">mean</dfn>, or <dfn class="terminology">expectation</dfn>) of <span class="process-math">\(X\)</span> is the weighted average of the values of <span class="process-math">\(X\text{,}\)</span> where the weights are the probabilities of <span class="process-math">\(X\)</span> taking each value:</div>
|
||||
<div class="displaymath process-math" id="def-discrete-expected-value-1-1-9">
|
||||
\begin{gather*}
|
||||
\E(X) = \sum_{i=1}^n x_i \Pr(X = x_i).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-expected-value-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 3.1.1"><a href="#def-discrete-expected-value" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div></article><article class="example example-like" id="example-EV-die"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
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</h4>
|
||||
<div class="para logical" id="example-EV-die-1-1">
|
||||
<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
|
||||
<div class="displaymath process-math" id="example-EV-die-1-1-4">
|
||||
\begin{align*}
|
||||
\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
|
||||
\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">For example, the expected value of a fair 6-sided die roll <span class="process-math">\(R\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="example-EV-die-1-1-6">
|
||||
\begin{gather*}
|
||||
\E(R) = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><article class="example example-like" id="example-EV-sum-2-dice"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="example-EV-sum-2-dice-1-1">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="example-EV-sum-2-dice-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k \cdot \Pr(S = k)\)</span></th>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(20/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(42/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(40/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(36/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">10</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">11</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(22/36\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">12</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para" id="example-EV-sum-2-dice-1-3">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><article class="example example-like" id="example-indicator-EV"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="example-indicator-EV-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="example-indicator-EV-1-1-4">
|
||||
\begin{gather*}
|
||||
\E(X) = 0 \cdot (1-p) + 1 \cdot p = p.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">That is, the expected value of an indicator random variable is the probability of the event that it indicates.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 3.1.5"><a href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div></article><div class="para" id="subsec-discrete-EV-7">Having found the expected value for one named distribution, we may be tempted to run through our other discrete distributions (binomial, geometric, and Poisson) and find nice formulas for their expected values. Unfortunately, both the geometric distribution and the Poisson distribution have infinitely many possible values, and the summation of infinitely many discrete terms is beyond the scope of this course. (A Calculus 2 course covering series would give the right tools to find expected value formulas with full justification for these distributions.)<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="subsec-discrete-EV-8">The binomial distribution takes only finitely many values, and we have <a href="sec-Discrete-RVs.html#fact-binomial-probability-formula" class="xref" data-knowl="./knowl/xref/fact-binomial-probability-formula.html" data-reveal-label="Reveal" data-close-label="Close" title="Fact 2.1.12: Binomial Probability Formula">Fact 2.1.12</a> that lets us generate a distribution table reasonably quickly, given the parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> Feel free to pick some parameter values (keep <span class="process-math">\(n\)</span> fairly small!) to try this calculation by hand, but we’ll use a powerful theorem a bit later to give a very nice general formula depending on <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Subsection 3.1.1: Discrete Expected Value"><a href="#subsec-discrete-EV" title="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value">🔗</a></div></section><section class="subsection" id="subsec-continuous-EV"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Continuous Expected Value</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-continuous-EV-2"><div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div>
|
||||
<div class="autopermalink" data-description="Subsection 3.1.2: Continuous Expected Value"><a href="#subsec-continuous-EV" title="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value">🔗</a></div></section><section class="subsection" id="subsec-linearity-EV"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Linearity of Expected Value</span>
|
||||
</h3>
|
||||
<div class="para" id="subsec-linearity-EV-2"><div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div>
|
||||
<div class="autopermalink" data-description="Subsection 3.1.3: Linearity of Expected Value"><a href="#subsec-linearity-EV" title="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value">🔗</a></div></section><section class="exercises" id="exercises-Expected-Value"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Expected-Value-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-1-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="exercises-Expected-Value-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.6: "><a href="#exercises-Expected-Value-1-1-2" title="Copy heading and permalink for Table 3.1.6: " aria-label="Copy heading and permalink for Table 3.1.6: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Exercise 3.1.4.1"><a href="#exercises-Expected-Value-1" title="Copy heading and permalink for Exercise 3.1.4.1" aria-label="Copy heading and permalink for Exercise 3.1.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-2-1-1">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables each taking the values 1, 2, 3, 4, 5. Write different distribution tables for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> so that they have the same expected value.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.2"><a href="#exercises-Expected-Value-2" title="Copy heading and permalink for Exercise 3.1.4.2" aria-label="Copy heading and permalink for Exercise 3.1.4.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-3-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads. If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\E(N)\text{?}\)</span> What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span> What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.3"><a href="#exercises-Expected-Value-3" title="Copy heading and permalink for Exercise 3.1.4.3" aria-label="Copy heading and permalink for Exercise 3.1.4.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-4-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\E(Y) = -2\text{,}\)</span> and <span class="process-math">\(\E(Z) = 1\text{,}\)</span> what is <span class="process-math">\(\E(4X + 5Y - Z + 3)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.4"><a href="#exercises-Expected-Value-4" title="Copy heading and permalink for Exercise 3.1.4.4" aria-label="Copy heading and permalink for Exercise 3.1.4.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-5-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.5"><a href="#exercises-Expected-Value-5" title="Copy heading and permalink for Exercise 3.1.4.5" aria-label="Copy heading and permalink for Exercise 3.1.4.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-6-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([-1, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{3x^2}{2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.6"><a href="#exercises-Expected-Value-6" title="Copy heading and permalink for Exercise 3.1.4.6" aria-label="Copy heading and permalink for Exercise 3.1.4.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-7-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.7"><a href="#exercises-Expected-Value-7" title="Copy heading and permalink for Exercise 3.1.4.7" aria-label="Copy heading and permalink for Exercise 3.1.4.7">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-8-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> In a previous problem, you found the value of <span class="process-math">\(k\text{.}\)</span> Now, find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.8"><a href="#exercises-Expected-Value-8" title="Copy heading and permalink for Exercise 3.1.4.8" aria-label="Copy heading and permalink for Exercise 3.1.4.8">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-9"><h4 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Expected-Value-9-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.1.4.9"><a href="#exercises-Expected-Value-9" title="Copy heading and permalink for Exercise 3.1.4.9" aria-label="Copy heading and permalink for Exercise 3.1.4.9">🔗</a></div></article><div class="autopermalink" data-description="Exercises 3.1.4"><a href="#exercises-Expected-Value" title="Copy heading and permalink for Exercises 3.1.4" aria-label="Copy heading and permalink for Exercises 3.1.4">🔗</a></div></section><div class="autopermalink" data-description="Section 3.1: Expected Value"><a href="#sec-Expected-Value" title="Copy heading and permalink for Section 3.1: Expected Value" aria-label="Copy heading and permalink for Section 3.1: Expected Value">🔗</a></div></section></div>
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Independent-Events"><h2 class="heading hide-type">
|
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Independent Events</span>
|
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</h2>
|
||||
<div class="para" id="sec-Independent-Events-2">The idea of conditional probability is that knowledge of one event can change our understanding of the probability of another. But it’s also important to understand when this is not the case.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Independent-Events-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="definition definition-like" id="def-independent-events"><h3 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="def-independent-events-1-1">Events <span class="process-math">\(A, B\)</span> are <dfn class="terminology">independent</dfn> if <span class="process-math">\(\Pr(A \mid B) = \Pr(A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-independent-events-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 1.4.1"><a href="#def-independent-events" title="Copy heading and permalink for Definition 1.4.1" aria-label="Copy heading and permalink for Definition 1.4.1">🔗</a></div></article><div class="para logical" id="sec-Independent-Events-4">
|
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<div class="para">The equation <span class="process-math">\(\Pr(A\mid B) = \Pr(A)\)</span> very concretely says: knowledge that the event <span class="process-math">\(B\)</span> has occurred does not change our understanding of <span class="process-math">\(\Pr(A)\text{.}\)</span> Assuming <span class="process-math">\(\Pr(A) \neq 0\text{,}\)</span> this is equivalent to <span class="process-math">\(\Pr(A\cap B) = \Pr(A)\Pr(B)\text{,}\)</span> since:</div>
|
||||
<div class="displaymath process-math" id="sec-Independent-Events-4-6">
|
||||
\begin{align*}
|
||||
\Pr(A\mid B) \amp = \Pr(A) \\
|
||||
\frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) \\
|
||||
\Pr(A\cap B) \amp = \Pr(A)\Pr(B)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">This latter form of the equation is often more useful in calculations.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Independent-Events-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<article class="example example-like" id="example-rolls-independent"><h3 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="example-rolls-independent-1-1">In <a href="sec-Conditional-Probability.html#example-rolls-conditional" class="xref" data-knowl="./knowl/xref/example-rolls-conditional.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 1.3.2">Example 1.3.2</a>, we considered an experiment in which we rolled a fair 6-sided die twice. We defined events <span class="process-math">\(A\)</span> that the sum of the rolls is at least 10 and <span class="process-math">\(B\)</span> that the first roll is a 6, and we found that <span class="process-math">\(\Pr(A \mid B) \neq \Pr(A)\text{,}\)</span> so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="example-rolls-independent-1-2">
|
||||
<div class="para">Now consider the event <span class="process-math">\(C\)</span> that the sum of the rolls is 7, which sounds very similar to the event <span class="process-math">\(A\text{.}\)</span> We have:</div>
|
||||
<div class="displaymath process-math" id="example-rolls-independent-1-2-3">
|
||||
\begin{align*}
|
||||
C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \\
|
||||
\Pr(C) \amp = \frac{6}{36} = \frac{1}{6} \\
|
||||
B\cap C \amp = \{(6, 1)\} \\
|
||||
\Pr(B\cap C) \amp = \frac{1}{36} \\
|
||||
\text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So, even though the descriptions of events <span class="process-math">\(A\)</span> and <span class="process-math">\(C\)</span> are very similar, the event <span class="process-math">\(C\)</span> is independent with <span class="process-math">\(B\text{,}\)</span> while <span class="process-math">\(A\)</span> is not.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Example 1.4.2"><a href="#example-rolls-independent" title="Copy heading and permalink for Example 1.4.2" aria-label="Copy heading and permalink for Example 1.4.2">🔗</a></div></article><section class="exercises" id="exercises-Independent-Events"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Independent-Events-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Independent-Events-1-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 1.4.1"><a href="#exercises-Independent-Events-1" title="Copy heading and permalink for Exercise 1.4.1" aria-label="Copy heading and permalink for Exercise 1.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Independent-Events-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Independent-Events-2-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.4.2"><a href="#exercises-Independent-Events-2" title="Copy heading and permalink for Exercise 1.4.2" aria-label="Copy heading and permalink for Exercise 1.4.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Independent-Events-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Independent-Events-3-1-1">Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{3, 4, 5\}\)</span> be events in the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> Create a probability distribution for <span class="process-math">\(\Omega\)</span> so that <span class="process-math">\(A, B\)</span> are independent.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 1.4.3"><a href="#exercises-Independent-Events-3" title="Copy heading and permalink for Exercise 1.4.3" aria-label="Copy heading and permalink for Exercise 1.4.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.4"><a href="#exercises-Independent-Events" title="Copy heading and permalink for Exercises 1.4" aria-label="Copy heading and permalink for Exercises 1.4">🔗</a></div></section><div class="autopermalink" data-description="Section 1.4: Independent Events"><a href="#sec-Independent-Events" title="Copy heading and permalink for Section 1.4: Independent Events" aria-label="Copy heading and permalink for Section 1.4: Independent Events">🔗</a></div></section></div>
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<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
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<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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</ul></nav></div>
|
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Probability"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Definition of Probability</span>
|
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</h2>
|
||||
<article class="definition definition-like" id="def-probability-distribution"><h3 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para logical" id="def-probability-distribution-1-1">
|
||||
<div class="para">A <dfn class="terminology">probability distribution</dfn> on a sample space <span class="process-math">\(\Omega\)</span> assigns probabilities to every event, satisfying the following conditions:</div>
|
||||
<ol class="decimal" id="def-probability-distribution-1-1-3">
|
||||
<li id="def-probability-distribution-1-1-3-1">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-1-1">
|
||||
<span class="process-math">\(\Pr(\Omega) = 1\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#def-probability-distribution-1-1-3-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-2">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-2-1">
|
||||
<span class="process-math">\(0 \leq \Pr(A) \leq 1\)</span> for any event <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
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<div class="autopermalink" data-description="Item 2"><a href="#def-probability-distribution-1-1-3-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||||
</li>
|
||||
<li id="def-probability-distribution-1-1-3-3">
|
||||
<div class="para" id="def-probability-distribution-1-1-3-3-1">If <span class="process-math">\(A \cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 3"><a href="#def-probability-distribution-1-1-3-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.2.1"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 1.2.1" aria-label="Copy heading and permalink for Definition 1.2.1">🔗</a></div></article><div class="para" id="sec-Probability-3">For small probability spaces (i.e., with finitely many outcomes in the sample space), we’ll usually assign probabilities to each individual outcome, and perhaps list them in a table. Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="sec-Probability-4"><h3 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="sec-Probability-4-1-1">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> The probability distribution (assuming a fair die) is shown below.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="sec-Probability-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 1.2.3: Distribution for a fair die"><a href="#sec-Probability-4-1-2" title="Copy heading and permalink for Table 1.2.3: Distribution for a fair die" aria-label="Copy heading and permalink for Table 1.2.3: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="sec-Probability-4-1-3">
|
||||
<div class="para">One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even. The probability of <span class="process-math">\(A\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="sec-Probability-4-1-3-3">
|
||||
\begin{gather*}
|
||||
\Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 1.2.2"><a href="#sec-Probability-4" title="Copy heading and permalink for Example 1.2.2" aria-label="Copy heading and permalink for Example 1.2.2">🔗</a></div></article><article class="definition definition-like" id="def-discrete-uniform-distribution"><h3 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="def-discrete-uniform-distribution-1-1">Let <span class="process-math">\(\Omega = \{x_1, x_2, \dotsc, x_n\}\)</span> be a sample space and <span class="process-math">\(A\subset \Omega\)</span> an event. We refer to the distribution in which <span class="process-math">\(\Pr(x_i) = \frac{1}{n}\)</span> for all <span class="process-math">\(i\)</span> as the <dfn class="terminology">uniform distribution</dfn>. In this case, it follows that <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-uniform-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.2.4"><a href="#def-discrete-uniform-distribution" title="Copy heading and permalink for Definition 1.2.4" aria-label="Copy heading and permalink for Definition 1.2.4">🔗</a></div></article><article class="example example-like" id="sec-Probability-6"><h3 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="sec-Probability-6-1-1">A person picks a random number from 1 to 10. What is the probability that they picked 3?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 1.2.5"><a href="#sec-Probability-6" title="Copy heading and permalink for Example 1.2.5" aria-label="Copy heading and permalink for Example 1.2.5">🔗</a></div></article><section class="exercises" id="exercises-Probability"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Probability-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Probability-1-1-1">Consider the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\)</span> with probability distribution below. Calculate the probabilities of <span class="process-math">\(A = \{1, 3, 7, 8\}\text{,}\)</span> <span class="process-math">\(B = \{2, 3, 6, 7\}\text{,}\)</span> <span class="process-math">\(A\cup B\text{,}\)</span> and <span class="process-math">\(A \cap B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="exercises-Probability-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 1.2.6: "><a href="#exercises-Probability-1-1-2" title="Copy heading and permalink for Table 1.2.6: " aria-label="Copy heading and permalink for Table 1.2.6: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
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<td class="c m b0 r0 l0 t0 lines">5</td>
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<td class="c m b0 r0 l0 t0 lines">6</td>
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<td class="c m b0 r0 l0 t0 lines">7</td>
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<td class="c m b0 r0 l0 t0 lines">0.05</td>
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<td class="c m b0 r0 l0 t0 lines">8</td>
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<td class="c m b0 r0 l0 t0 lines">0.1</td>
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<td class="c m b0 r0 l0 t0 lines">9</td>
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<td class="c m b0 r0 l0 t0 lines">0.1</td>
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|
||||
</table></div></figure> <div class="autopermalink" data-description="Exercise 1.2.1"><a href="#exercises-Probability-1" title="Copy heading and permalink for Exercise 1.2.1" aria-label="Copy heading and permalink for Exercise 1.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="introduction" id="exercises-Probability-2-1"><div class="para" id="exercises-Probability-2-1-1">Suppose we flip a coin two times. Answer the questions below. What about three flips? What about four flips?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div>
|
||||
<article class="task exercise-like" id="exercises-Probability-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||||
<div class="para" id="exercises-Probability-2-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.2.a"><a href="#exercises-Probability-2-2" title="Copy heading and permalink for Task 1.2.2.a" aria-label="Copy heading and permalink for Task 1.2.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||||
<div class="para" id="exercises-Probability-2-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the coin is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.2.b"><a href="#exercises-Probability-2-3" title="Copy heading and permalink for Task 1.2.2.b" aria-label="Copy heading and permalink for Task 1.2.2.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
|
||||
<div class="para" id="exercises-Probability-2-4-1-1">Make a probability distribution table assuming the coin comes up heads with probability 0.3.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.2.c"><a href="#exercises-Probability-2-4" title="Copy heading and permalink for Task 1.2.2.c" aria-label="Copy heading and permalink for Task 1.2.2.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.2"><a href="#exercises-Probability-2" title="Copy heading and permalink for Exercise 1.2.2" aria-label="Copy heading and permalink for Exercise 1.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="introduction" id="exercises-Probability-3-1"><div class="para" id="exercises-Probability-3-1-1">Suppose we roll a die two times. Answer the questions below.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div>
|
||||
<article class="task exercise-like" id="exercises-Probability-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||||
<div class="para" id="exercises-Probability-3-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.3.a"><a href="#exercises-Probability-3-2" title="Copy heading and permalink for Task 1.2.3.a" aria-label="Copy heading and permalink for Task 1.2.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||||
<div class="para" id="exercises-Probability-3-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the die is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.3.b"><a href="#exercises-Probability-3-3" title="Copy heading and permalink for Task 1.2.3.b" aria-label="Copy heading and permalink for Task 1.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
|
||||
<div class="para" id="exercises-Probability-3-4-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is higher than the first, and let <span class="process-math">\(B\)</span> be the event that the first roll is even. Find <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> and <span class="process-math">\(\Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="autopermalink" data-description="Task 1.2.3.c"><a href="#exercises-Probability-3-4" title="Copy heading and permalink for Task 1.2.3.c" aria-label="Copy heading and permalink for Task 1.2.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.3"><a href="#exercises-Probability-3" title="Copy heading and permalink for Exercise 1.2.3" aria-label="Copy heading and permalink for Exercise 1.2.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Probability-4-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.2.4"><a href="#exercises-Probability-4" title="Copy heading and permalink for Exercise 1.2.4" aria-label="Copy heading and permalink for Exercise 1.2.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Probability-5-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.2.5"><a href="#exercises-Probability-5" title="Copy heading and permalink for Exercise 1.2.5" aria-label="Copy heading and permalink for Exercise 1.2.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Probability-6-1-1">A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> find the probability of the toxin molecule leaving the cell during the <span class="process-math">\(n\)</span>th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.2.6"><a href="#exercises-Probability-6" title="Copy heading and permalink for Exercise 1.2.6" aria-label="Copy heading and permalink for Exercise 1.2.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Probability-7-1-1">Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> and for each value of <span class="process-math">\(0\leq k \leq n\text{,}\)</span> find the probability that exactly <span class="process-math">\(k\)</span> toxin molecules remain in the cell after the <span class="process-math">\(n\)</span>th minute.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.2.7"><a href="#exercises-Probability-7" title="Copy heading and permalink for Exercise 1.2.7" aria-label="Copy heading and permalink for Exercise 1.2.7">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.2"><a href="#exercises-Probability" title="Copy heading and permalink for Exercises 1.2" aria-label="Copy heading and permalink for Exercises 1.2">🔗</a></div></section><div class="autopermalink" data-description="Section 1.2: Definition of Probability"><a href="#sec-Probability" title="Copy heading and permalink for Section 1.2: Definition of Probability" aria-label="Copy heading and permalink for Section 1.2: Definition of Probability">🔗</a></div></section></div>
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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Set Theory</span>
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<article class="definition definition-like" id="def-sample-space"><h3 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para" id="def-sample-space-1-1">The <dfn class="terminology">sample space</dfn>, often denoted <span class="process-math">\(\Omega\text{,}\)</span> is the set of all possible results of an experiment. A single result is called an <dfn class="terminology">outcome</dfn>, while a collection of results is called an <dfn class="terminology">event</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 1.1.1"><a href="#def-sample-space" title="Copy heading and permalink for Definition 1.1.1" aria-label="Copy heading and permalink for Definition 1.1.1">🔗</a></div></article><article class="example example-like" id="sec-Set-Theory-3"><h3 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h3>
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||||
<div class="para" id="sec-Set-Theory-3-1-1">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Set-Theory-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <div class="para" id="sec-Set-Theory-3-1-2">What would the sample space look like if we roll the die two times and recorded the results?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Set-Theory-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="sec-Set-Theory-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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\begin{align*}
|
||||
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
|
||||
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
|
||||
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
|
||||
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
|
||||
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
|
||||
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
|
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\end{align*}
|
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<div class="autopermalink" data-description="Paragraph"><a href="#sec-Set-Theory-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 1.1.2.1"><a href="#sec-Set-Theory-3-2" title="Copy heading and permalink for Answer 1.1.2.1" aria-label="Copy heading and permalink for Answer 1.1.2.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Example 1.1.2"><a href="#sec-Set-Theory-3" title="Copy heading and permalink for Example 1.1.2" aria-label="Copy heading and permalink for Example 1.1.2">🔗</a></div></article><article class="definition definition-like" id="def-subset"><h3 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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||||
<div class="para" id="def-subset-1-1">Let <span class="process-math">\(A\)</span> be a set. The symbol <span class="process-math">\(\in\)</span> means "is an element of", as in <span class="process-math">\(a \in A\text{.}\)</span> Given another set <span class="process-math">\(B\text{,}\)</span> we say <span class="process-math">\(A\)</span> is a <dfn class="terminology">subset</dfn> of <span class="process-math">\(B\text{,}\)</span> written <span class="process-math">\(A\subset B\text{,}\)</span> to mean that every element of the set <span class="process-math">\(A\)</span> is also an element of the set <span class="process-math">\(B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
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||||
<div class="autopermalink" data-description="Definition 1.1.3"><a href="#def-subset" title="Copy heading and permalink for Definition 1.1.3" aria-label="Copy heading and permalink for Definition 1.1.3">🔗</a></div></article><article class="definition definition-like" id="def-set-operations"><h3 class="heading">
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||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h3>
|
||||
<div class="para logical" id="def-set-operations-1-1">
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||||
<div class="para">Consider sets <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> each contained inside <span class="process-math">\(\Omega\text{.}\)</span> We can combine sets in a variety of ways:</div>
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<dl class="description-list">
|
||||
<dt id="def-set-operations-1-1-4-1">Union<div class="autopermalink" data-description="Item: Union"><a href="#def-set-operations-1-1-4-1" title="Copy heading and permalink for Item: Union" aria-label="Copy heading and permalink for Item: Union">🔗</a></div>
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||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-1-2">The <dfn class="terminology">union</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-2">Intersection<div class="autopermalink" data-description="Item: Intersection"><a href="#def-set-operations-1-1-4-2" title="Copy heading and permalink for Item: Intersection" aria-label="Copy heading and permalink for Item: Intersection">🔗</a></div>
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||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-2-2">The <dfn class="terminology">intersection</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-3">Difference<div class="autopermalink" data-description="Item: Difference"><a href="#def-set-operations-1-1-4-3" title="Copy heading and permalink for Item: Difference" aria-label="Copy heading and permalink for Item: Difference">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-3-2">The <dfn class="terminology">set difference</dfn> <span class="process-math">\(A-B\)</span> is the set <span class="process-math">\(A - B = \{x \mid x \in A \text{ and } x \notin B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-4">Complement<div class="autopermalink" data-description="Item: Complement"><a href="#def-set-operations-1-1-4-4" title="Copy heading and permalink for Item: Complement" aria-label="Copy heading and permalink for Item: Complement">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-4-2">The <dfn class="terminology">complement</dfn> of <span class="process-math">\(A\)</span> is the set <span class="process-math">\(A^c = \{x \in \Omega \mid x \notin A\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
<dt id="def-set-operations-1-1-4-5">Empty Set<div class="autopermalink" data-description="Item: Empty Set"><a href="#def-set-operations-1-1-4-5" title="Copy heading and permalink for Item: Empty Set" aria-label="Copy heading and permalink for Item: Empty Set">🔗</a></div>
|
||||
</dt>
|
||||
<dd><div class="para" id="def-set-operations-1-1-4-5-2">The <dfn class="terminology">empty set</dfn>, usually written <span class="process-math">\(\emptyset\)</span> or <span class="process-math">\(\{\}\text{,}\)</span> is the set which contains no elements.<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-5-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></dd>
|
||||
</dl>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.1.4"><a href="#def-set-operations" title="Copy heading and permalink for Definition 1.1.4" aria-label="Copy heading and permalink for Definition 1.1.4">🔗</a></div></article><div class="para" id="sec-Set-Theory-6">It’s useful sometimes to draw pictures representing sets...<div class="autopermalink" data-description="Paragraph"><a href="#sec-Set-Theory-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-disjoint"><h3 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="def-disjoint-1-1">Two sets <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are <dfn class="terminology">disjoint</dfn> if <span class="process-math">\(A \cap B = \emptyset\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-disjoint-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.1.5"><a href="#def-disjoint" title="Copy heading and permalink for Definition 1.1.5" aria-label="Copy heading and permalink for Definition 1.1.5">🔗</a></div></article><article class="definition definition-like" id="def-cardinality"><h3 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="def-cardinality-1-1">Given a finite set <span class="process-math">\(A\text{,}\)</span> the <dfn class="terminology">cardinality</dfn> of <span class="process-math">\(A\text{,}\)</span> written <span class="process-math">\(|A|\text{,}\)</span> is the number of elements in <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-cardinality-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 1.1.6"><a href="#def-cardinality" title="Copy heading and permalink for Definition 1.1.6" aria-label="Copy heading and permalink for Definition 1.1.6">🔗</a></div></article><section class="exercises" id="exercises-Set-Theory"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||||
</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Set-Theory-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="introduction" id="exercises-Set-Theory-1-1"><div class="para" id="exercises-Set-Theory-1-1-1">Consider the sets <span class="process-math">\(A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\text{,}\)</span> <span class="process-math">\(B = \{2, 4, 9, 10, 12, 14, 19\}\text{,}\)</span> and <span class="process-math">\(C = \{9, 10, 11, 14, 16, 17, 20\}\text{,}\)</span> which are all subsets of <span class="process-math">\(\Omega = \{1, 2, 3, \dotsc, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></div>
|
||||
<article class="task exercise-like" id="exercises-Set-Theory-1-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||||
<div class="para" id="exercises-Set-Theory-1-2-1-1">Find <span class="process-math">\(A - (B \cap C)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-Set-Theory-1-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-Set-Theory-1-2-2-1">
|
||||
<span class="process-math">\(\{1, 2, 3, 4, 5, 6, 7, 8\}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 1.1.1.a.1"><a href="#exercises-Set-Theory-1-2-2" title="Copy heading and permalink for Answer 1.1.1.a.1" aria-label="Copy heading and permalink for Answer 1.1.1.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.1.1.a"><a href="#exercises-Set-Theory-1-2" title="Copy heading and permalink for Task 1.1.1.a" aria-label="Copy heading and permalink for Task 1.1.1.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Set-Theory-1-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||||
<div class="para" id="exercises-Set-Theory-1-3-1-1">Find <span class="process-math">\(|A|\text{,}\)</span> <span class="process-math">\(|B|\text{,}\)</span> <span class="process-math">\(|C|\text{,}\)</span> <span class="process-math">\(|A\cup B|\text{,}\)</span> <span class="process-math">\(|A \cap B|\text{,}\)</span> <span class="process-math">\(|B\cap C|\text{,}\)</span> <span class="process-math">\(|A\cap C|\text{,}\)</span> and <span class="process-math">\(|A\cup B\cup C|\text{.}\)</span> Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-Set-Theory-1-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-Set-Theory-1-3-2-1">
|
||||
<span class="process-math">\(|A| = 10\text{,}\)</span> <span class="process-math">\(|B| = 7\text{,}\)</span> <span class="process-math">\(|C| = 7\text{,}\)</span> <span class="process-math">\(|A \cup B| = 13\text{,}\)</span> <span class="process-math">\(|A \cap B| = 4\text{,}\)</span> <span class="process-math">\(|B\cap C| = 3\text{,}\)</span> <span class="process-math">\(|A\cap C| = 2\text{,}\)</span> <span class="process-math">\(|A\cup B\cup C| = 17\text{.}\)</span> In particular, note that <span class="process-math">\(|A\cup B| = 13 \neq 10 + 7 = |A| + |B|\text{,}\)</span> so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 1.1.1.b.1"><a href="#exercises-Set-Theory-1-3-2" title="Copy heading and permalink for Answer 1.1.1.b.1" aria-label="Copy heading and permalink for Answer 1.1.1.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.1.1.b"><a href="#exercises-Set-Theory-1-3" title="Copy heading and permalink for Task 1.1.1.b" aria-label="Copy heading and permalink for Task 1.1.1.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Set-Theory-1-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
|
||||
<div class="para" id="exercises-Set-Theory-1-4-1-1">Find <span class="process-math">\(A^c\)</span> and <span class="process-math">\((A\cup B)^c\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-Set-Theory-1-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-Set-Theory-1-4-2-1">
|
||||
<span class="process-math">\(A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\text{,}\)</span> <span class="process-math">\((A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 1.1.1.c.1"><a href="#exercises-Set-Theory-1-4-2" title="Copy heading and permalink for Answer 1.1.1.c.1" aria-label="Copy heading and permalink for Answer 1.1.1.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.1.1.c"><a href="#exercises-Set-Theory-1-4" title="Copy heading and permalink for Task 1.1.1.c" aria-label="Copy heading and permalink for Task 1.1.1.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.1.1"><a href="#exercises-Set-Theory-1" title="Copy heading and permalink for Exercise 1.1.1" aria-label="Copy heading and permalink for Exercise 1.1.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Set-Theory-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Set-Theory-2-1-1">Suppose we roll a 6-sided die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <span class="process-math">\((4, 2)\)</span>---rolling a 4 and then a 2.]<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.1.2"><a href="#exercises-Set-Theory-2" title="Copy heading and permalink for Exercise 1.1.2" aria-label="Copy heading and permalink for Exercise 1.1.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Set-Theory-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Set-Theory-3-1-1">Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.1.3"><a href="#exercises-Set-Theory-3" title="Copy heading and permalink for Exercise 1.1.3" aria-label="Copy heading and permalink for Exercise 1.1.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Set-Theory-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Set-Theory-4-1-1">If we roll a 6-sided die ten times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Set-Theory-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 1.1.4"><a href="#exercises-Set-Theory-4" title="Copy heading and permalink for Exercise 1.1.4" aria-label="Copy heading and permalink for Exercise 1.1.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.1"><a href="#exercises-Set-Theory" title="Copy heading and permalink for Exercises 1.1" aria-label="Copy heading and permalink for Exercises 1.1">🔗</a></div></section><div class="autopermalink" data-description="Section 1.1: Set Theory"><a href="#sec-Set-Theory" title="Copy heading and permalink for Section 1.1: Set Theory" aria-label="Copy heading and permalink for Section 1.1: Set Theory">🔗</a></div></section></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
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</li>
|
||||
<li class="toc-item toc-chapter contains-active">
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||||
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
|
||||
<ul class="structural toc-item-list contains-active">
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<li class="toc-item toc-section">
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||||
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
|
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</ul>
|
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</li>
|
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<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
|
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<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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<li class="toc-item toc-section">
|
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
|
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
|
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</ul>
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
|
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</ul>
|
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<li class="toc-item toc-chapter">
|
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
|
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
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</ul>
|
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
|
||||
<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
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<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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</ul></nav></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Variance"><h2 class="heading hide-type">
|
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Variance</span>
|
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</h2>
|
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<div class="para" id="sec-Variance-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<section class="exercises" id="exercises-Variance"><h3 class="heading hide-type">
|
||||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
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</h3>
|
||||
<article class="exercise exercise-like" id="exercises-Variance-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-1-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="exercises-Variance-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.2.1: "><a href="#exercises-Variance-1-1-2" title="Copy heading and permalink for Table 3.2.1: " aria-label="Copy heading and permalink for Table 3.2.1: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
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<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Exercise 3.2.1"><a href="#exercises-Variance-1" title="Copy heading and permalink for Exercise 3.2.1" aria-label="Copy heading and permalink for Exercise 3.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-2-1-1">Let <span class="process-math">\(X\)</span> be a random variable taking the values 1, 2, 3, 4, 5. Write a distribution table for <span class="process-math">\(X\text{,}\)</span> then use your table to write a distribution for <span class="process-math">\(X^2\text{.}\)</span> Then, find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.2"><a href="#exercises-Variance-2" title="Copy heading and permalink for Exercise 3.2.2" aria-label="Copy heading and permalink for Exercise 3.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-3-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads. If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\Var(N)\text{?}\)</span> What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span> What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.3"><a href="#exercises-Variance-3" title="Copy heading and permalink for Exercise 3.2.3" aria-label="Copy heading and permalink for Exercise 3.2.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-4-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\Var(X) = 2\text{,}\)</span> what is <span class="process-math">\(\E(X^2)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.4"><a href="#exercises-Variance-4" title="Copy heading and permalink for Exercise 3.2.4" aria-label="Copy heading and permalink for Exercise 3.2.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-5-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.5"><a href="#exercises-Variance-5" title="Copy heading and permalink for Exercise 3.2.5" aria-label="Copy heading and permalink for Exercise 3.2.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-6-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([-1, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{3x^2}{2}\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.6"><a href="#exercises-Variance-6" title="Copy heading and permalink for Exercise 3.2.6" aria-label="Copy heading and permalink for Exercise 3.2.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-7-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.7"><a href="#exercises-Variance-7" title="Copy heading and permalink for Exercise 3.2.7" aria-label="Copy heading and permalink for Exercise 3.2.7">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-8-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> In a previous problem, you found the value of <span class="process-math">\(k\text{.}\)</span> Now, find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.8"><a href="#exercises-Variance-8" title="Copy heading and permalink for Exercise 3.2.8" aria-label="Copy heading and permalink for Exercise 3.2.8">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-9"><h4 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Variance-9-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Exercise 3.2.9"><a href="#exercises-Variance-9" title="Copy heading and permalink for Exercise 3.2.9" aria-label="Copy heading and permalink for Exercise 3.2.9">🔗</a></div></article><div class="autopermalink" data-description="Exercises 3.2"><a href="#exercises-Variance" title="Copy heading and permalink for Exercises 3.2" aria-label="Copy heading and permalink for Exercises 3.2">🔗</a></div></section><div class="autopermalink" data-description="Section 3.2: Variance"><a href="#sec-Variance" title="Copy heading and permalink for Section 3.2: Variance" aria-label="Copy heading and permalink for Section 3.2: Variance">🔗</a></div></section></div>
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