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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">
<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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@@ -208,7 +181,7 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-section">
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@@ -218,16 +191,16 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Joint-Distributions" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</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-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 id="ptx-toc-group-ch-Expected-Value" 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">
<ul id="ptx-toc-group-sec-Expected-Value" 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>
@@ -236,91 +209,98 @@ eBookConfig.enable_chatcodes = false;
</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>
<ul id="ptx-toc-group-sec-Variance" 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>
<ul id="ptx-toc-group-sec-Covariance" 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>
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<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Confidence-Intervals-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Likelihood.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Likelihood</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Likelihood.html#sec-Likelihood-20" class="internal"><span class="codenumber">4.1</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Likelihood" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Likelihood.html#sec-Likelihood-20" class="internal"><span class="codenumber">4.1</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-CLT.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Central Limit Theorem</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-CLT" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</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-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Confidence-Intervals" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
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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 id="ptx-toc-group-ch-Hypothesis-Testing" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Hypothesis-Testing-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
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<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>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-One-Sample-Tests" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</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-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Two-Sample-Tests" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<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>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Power" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<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>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Chi-Squared" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
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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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<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>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Correlation" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</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-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Linear-Regression" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
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<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
<ul id="ptx-toc-group-backmatter-2" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
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<li class="toc-item toc-appendix">
<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
<ul id="ptx-toc-group-backmatter-3" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
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<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-4.html" class="internal"><span class="title">Colophon</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">
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
<section class="section" id="sec-Independent-Events">
<h1 class="heading hide-type">
<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>
</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>
</h1>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Independent-Events-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-independent-events"><h3 class="heading">
<article class="definition definition-like" id="def-independent-events"><h2 class="heading">
<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>
</h2>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-independent-events-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 1.4.1"><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">
<div class="autopermalink" aria-hidden="true" data-description="Definition 1.4.1"><a tabindex="-1" 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">
<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*}
@@ -330,13 +310,14 @@ eBookConfig.enable_chatcodes = false;
\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>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Independent-Events-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-rolls-independent"><h3 class="heading">
<article class="example example-like" id="example-rolls-independent"><h2 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">
</h2>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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*}
@@ -348,15 +329,20 @@ B\cap C \amp = \{(6, 1)\} \\
\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>
<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>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-rolls-independent-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 1.4.2"><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">
<div class="autopermalink" aria-hidden="true" data-description="Example 1.4.2"><a tabindex="-1" 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">
<h2 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>
</h2>
<article class="exercise exercise-like" id="exercises-Independent-Events-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Independent-Events-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Independent-Events-1-2" 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="solutions">
<details id="exercises-Independent-Events-1-2" 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 logical" id="exercises-Independent-Events-1-2-1">
<div class="displaymath process-math" id="exercises-Independent-Events-1-2-1-1">
\begin{align*}
@@ -378,14 +364,19 @@ A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.</div>
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</div>
<div class="autopermalink" data-description="Answer 1.4.1.1"><a href="#exercises-Independent-Events-1-2" title="Copy heading and permalink for Answer 1.4.1.1" aria-label="Copy heading and permalink for Answer 1.4.1.1">🔗</a></div>
</div></details></div>
<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></details>
</div>
<div class="solutions"><details id="exercises-Independent-Events-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="autopermalink" aria-hidden="true" data-description="Exercise 1.4.1"><a tabindex="-1" 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"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Independent-Events-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-Independent-Events-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 logical" id="exercises-Independent-Events-2-2-1">
<div class="displaymath process-math" id="exercises-Independent-Events-2-2-1-1">
\begin{align*}
@@ -401,15 +392,20 @@ A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<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>
<div class="solutions"><details id="exercises-Independent-Events-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">
<figure class="table table-like" id="exercises-Independent-Events-3-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 1.4.3: Example Distribution"><a href="#exercises-Independent-Events-3-2-1" title="Copy heading and permalink for Table 1.4.3: Example Distribution" aria-label="Copy heading and permalink for Table 1.4.3: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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</article>
<article class="exercise exercise-like" id="exercises-Independent-Events-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Independent-Events-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-Independent-Events-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">
<figure class="table table-like" id="exercises-Independent-Events-3-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" aria-hidden="true" data-description="Table 1.4.3: Example Distribution"><a tabindex="-1" href="#exercises-Independent-Events-3-2-1" title="Copy heading and permalink for Table 1.4.3: Example Distribution" aria-label="Copy heading and permalink for Table 1.4.3: Example Distribution">🔗</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>
@@ -438,7 +434,9 @@ A\cap B \amp = \{HHH, HHT\}
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure><div class="para logical" id="exercises-Independent-Events-3-2-2">
</table></div>
</figure>
<div class="para logical" id="exercises-Independent-Events-3-2-2">
<div class="para">Now <span class="process-math">\(\Pr(A) = 0.5\text{,}\)</span> <span class="process-math">\(\Pr(B) = 0.4\text{,}\)</span> and</div>
<div class="displaymath process-math" id="exercises-Independent-Events-3-2-2-3">
\begin{gather*}
@@ -446,11 +444,16 @@ A\cap B \amp = \{HHH, HHT\}
\end{gather*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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</section><div class="autopermalink" aria-hidden="true" data-description="Section 1.4: Independent Events"><a tabindex="-1" 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>
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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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