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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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<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-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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<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>
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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-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="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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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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<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>
|
||
<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 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 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>
|
||
<ul id="ptx-toc-group-ch-Confidence-Intervals" class="structural toc-item-list">
|
||
<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 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 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 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>
|
||
</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 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>
|
||
<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>
|
||
<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>
|
||
</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>
|
||
<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>
|
||
</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>
|
||
<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>
|
||
</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>
|
||
<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>
|
||
</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 id="ptx-toc-group-ch-Linear-Regression" class="structural toc-item-list">
|
||
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Linear-Regression-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-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
|
||
<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>
|
||
</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>
|
||
<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>
|
||
</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>
|
||
<ul id="ptx-toc-group-backmatter" class="structural toc-item-list">
|
||
<li class="toc-item toc-appendix">
|
||
<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 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>
|
||
</li>
|
||
<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 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>
|
||
</li>
|
||
<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>
|
||
</ul>
|
||
</li>
|
||
</ul></nav></div>
|
||
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
|
||
<section class="section" id="sec-Continuous-RVs">
|
||
<h1 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>
|
||
</h1>
|
||
<section class="subsection" id="subsec-pdfs">
|
||
<h2 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>
|
||
</h2>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item 1"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item 2"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Definition 2.2.1"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Definition 2.2.2"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Example 2.2.3"><a tabindex="-1" 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" aria-hidden="true" data-description="Subsection 2.2.1: Probability Density Functions"><a tabindex="-1" 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">
|
||
<h2 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>
|
||
</h2>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Definition 2.2.4"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Example 2.2.5"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Subsection 2.2.2: Cumulative Distribution Functions"><a tabindex="-1" 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">
|
||
<h2 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>
|
||
</h2>
|
||
<article class="definition definition-like" id="def-exponential-distribution"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Definition 2.2.6"><a tabindex="-1" 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"><h3 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>
|
||
</h3>
|
||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Fact 2.2.7: Exponential Density Function"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item 1"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item 2"><a tabindex="-1" 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" aria-hidden="true" data-description="Item: Binomial"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item: Geometric"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item: Poisson"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Item: Exponential"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Subsection 2.2.3: The Exponential Distribution"><a tabindex="-1" 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">
|
||
<h2 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>
|
||
</h2>
|
||
<article class="definition definition-like" id="def-normal-distribution"><h3 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>
|
||
</h3>
|
||
<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>. We’ll write <span class="process-math">\(X \sim \Norm(\mu, \sigma^2)\text{.}\)</span>
|
||
</div>
|
||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="para" id="def-normal-distribution-1-2">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>. We’ll use the notation <span class="process-math">\(\phi(x)\)</span> for the standard normal density function, and <span class="process-math">\(\Phi(x)\)</span> for the standard normal cumulative distribution function.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-normal-distribution-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" aria-hidden="true" data-description="Definition 2.2.8"><a tabindex="-1" 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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#subsec-normal-distribution-3" 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" aria-hidden="true" data-description="Subsection 2.2.4: Normal Distribution"><a tabindex="-1" 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>
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</section>
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<section class="exercises" id="exercises-Continuous-RVs">
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<h2 class="heading hide-type">
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<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>
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</h2>
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<article class="exercise exercise-like" id="exercises-Continuous-RVs-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<div class="introduction" id="exercises-Continuous-RVs-1-1">
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<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><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-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>
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</div>
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<article class="task exercise-like" id="exercises-Continuous-RVs-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="exercises-Continuous-RVs-1-2-1-1">What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions">
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<details id="exercises-Continuous-RVs-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>
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<div class="answer solution-like knowl__content">
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<div class="para" id="exercises-Continuous-RVs-1-2-2-1">
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<span class="process-math">\(k = \frac{6}{17}.\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Answer 2.2.5.1.a.1"><a tabindex="-1" href="#exercises-Continuous-RVs-1-2-2" title="Copy heading and permalink for Answer 2.2.5.1.a.1" aria-label="Copy heading and permalink for Answer 2.2.5.1.a.1">🔗</a></div>
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</div></details>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Task 2.2.5.1.a"><a tabindex="-1" href="#exercises-Continuous-RVs-1-2" title="Copy heading and permalink for Task 2.2.5.1.a" aria-label="Copy heading and permalink for Task 2.2.5.1.a">🔗</a></div>
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</article>
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<article class="task exercise-like" id="exercises-Continuous-RVs-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
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||
<div class="para" id="exercises-Continuous-RVs-1-3-1-1">Find <span class="process-math">\(\Pr(2 \leq X \leq 3).\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-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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</div>
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<div class="solutions">
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<details id="exercises-Continuous-RVs-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>
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<div class="answer solution-like knowl__content">
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<div class="para" id="exercises-Continuous-RVs-1-3-2-1">
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<span class="process-math">\(\Pr(2 \leq X \leq 3) \approx 0.325\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-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" aria-hidden="true" data-description="Answer 2.2.5.1.b.1"><a tabindex="-1" href="#exercises-Continuous-RVs-1-3-2" title="Copy heading and permalink for Answer 2.2.5.1.b.1" aria-label="Copy heading and permalink for Answer 2.2.5.1.b.1">🔗</a></div>
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</div></details>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Task 2.2.5.1.b"><a tabindex="-1" href="#exercises-Continuous-RVs-1-3" title="Copy heading and permalink for Task 2.2.5.1.b" aria-label="Copy heading and permalink for Task 2.2.5.1.b">🔗</a></div>
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</article><div class="autopermalink" aria-hidden="true" data-description="Exercise 2.2.5.1"><a tabindex="-1" 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>
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</article>
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<article class="exercise exercise-like" id="exercises-Continuous-RVs-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-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>
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||
<div class="solutions">
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<details id="exercises-Continuous-RVs-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>
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||
<div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Continuous-RVs-2-2-1">
|
||
<span class="process-math">\(\frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}\text{.}\)</span> <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479.\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-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>
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<div class="autopermalink" aria-hidden="true" data-description="Answer 2.2.5.2.1"><a tabindex="-1" href="#exercises-Continuous-RVs-2-2" title="Copy heading and permalink for Answer 2.2.5.2.1" aria-label="Copy heading and permalink for Answer 2.2.5.2.1">🔗</a></div>
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</div></details>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Exercise 2.2.5.2"><a tabindex="-1" 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>
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</article>
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<article class="exercise exercise-like" id="exercises-Continuous-RVs-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-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>
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<div class="solutions">
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<details id="exercises-Continuous-RVs-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-Continuous-RVs-3-2-1">
|
||
<span class="process-math">\(f(x) = x^2 - 2x.\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-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" aria-hidden="true" data-description="Answer 2.2.5.3.1"><a tabindex="-1" href="#exercises-Continuous-RVs-3-2" title="Copy heading and permalink for Answer 2.2.5.3.1" aria-label="Copy heading and permalink for Answer 2.2.5.3.1">🔗</a></div>
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</div></details>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Exercise 2.2.5.3"><a tabindex="-1" 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>
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</article>
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<article class="exercise exercise-like" id="exercises-Continuous-RVs-4"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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||
<div class="para" id="exercises-Continuous-RVs-4-1-1">Let <span class="process-math">\(T \sim \Exp(3)\text{.}\)</span> Find <span class="process-math">\(\Pr(1 \leq T \leq 3)\)</span> and <span class="process-math">\(\Pr(T \geq 0.5)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-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>
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<div class="solutions">
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<details id="exercises-Continuous-RVs-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>
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||
<div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Continuous-RVs-4-2-1">
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||
<span class="process-math">\(\Pr(1 \leq T \leq 3) \approx 0.050\text{.}\)</span> <span class="process-math">\(\Pr(T \geq 0.5) \approx 0.223\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Continuous-RVs-4-2-1" 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" aria-hidden="true" data-description="Answer 2.2.5.4.1"><a tabindex="-1" href="#exercises-Continuous-RVs-4-2" title="Copy heading and permalink for Answer 2.2.5.4.1" aria-label="Copy heading and permalink for Answer 2.2.5.4.1">🔗</a></div>
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</div></details>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Exercise 2.2.5.4"><a tabindex="-1" 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>
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</article><div class="autopermalink" aria-hidden="true" data-description="Exercises 2.2.5"><a tabindex="-1" 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>
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</section><div class="autopermalink" aria-hidden="true" data-description="Section 2.2: Continuous Random Variables"><a tabindex="-1" 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>
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