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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<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-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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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>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<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>
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<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>
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<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>
|
||
</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>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
|
||
<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>
|
||
</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 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 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 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 class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
|
||
<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>
|
||
</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 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>
|
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<li class="toc-item toc-backmatter">
|
||
<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
|
||
<ul 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 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-colophon"><div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="title">Colophon</span></a></div></li>
|
||
</ul>
|
||
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|
||
</ul></nav></div>
|
||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Expected-Value"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Expected Value</span>
|
||
</h2>
|
||
<section class="subsection" id="subsec-discrete-EV"><h3 class="heading hide-type">
|
||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Discrete Expected Value</span>
|
||
</h3>
|
||
<div class="para" id="subsec-discrete-EV-2">A full probability distribution (for a discrete random variable) or a density function (for a continuous random variable) carry all of the probability information. Often, we seek <dfn class="terminology">statistics</dfn>, i.e., numbers that summarize a distribution in some way.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-discrete-EV"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="def-discrete-EV-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> (also called <dfn class="terminology">mean</dfn>, or <dfn class="terminology">expectation</dfn>) of <span class="process-math">\(X\)</span> is the weighted average of the values of <span class="process-math">\(X\text{,}\)</span> where the weights are the probabilities of <span class="process-math">\(X\)</span> taking each value:</div>
|
||
<div class="displaymath process-math" id="def-discrete-EV-1-1-9">
|
||
\begin{gather*}
|
||
\E(X) = \sum_{i=1}^n x_i \Pr(X = x_i).
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 3.1.1"><a href="#def-discrete-EV" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div></article><article class="example example-like" id="example-EV-die"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="example-EV-die-1-1">
|
||
<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
|
||
<div class="displaymath process-math" id="example-EV-die-1-1-4">
|
||
\begin{align*}
|
||
\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
|
||
\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">For example, the expected value of a fair 6-sided die roll <span class="process-math">\(R\)</span> is:</div>
|
||
<div class="displaymath process-math" id="example-EV-die-1-1-6">
|
||
\begin{align*}
|
||
\E(R) \amp = 1 \left(\frac{1}{6}\right) + 2 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{6}\right) + 4 \left(\frac{1}{6}\right) + 5 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{6}\right) \\
|
||
\amp = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><article class="example example-like" id="example-EV-sum-2-dice"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="example-EV-sum-2-dice-1-1">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="example-EV-sum-2-dice-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr class="header-horizontal">
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k \cdot \Pr(S = k)\)</span></th>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(20/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(42/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(40/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(36/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">10</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">11</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(22/36\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">12</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
|
||
</tr>
|
||
</table></div></figure> <div class="para" id="example-EV-sum-2-dice-1-3">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><article class="example example-like" id="example-indicator-EV"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="example-indicator-EV-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> Then:</div>
|
||
<div class="displaymath process-math" id="example-indicator-EV-1-1-4">
|
||
\begin{gather*}
|
||
\E(X) = 0 \cdot (1-p) + 1 \cdot p = p.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">That is, the expected value of an indicator random variable is the probability of the event that it indicates.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.5"><a href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div></article><div class="para" id="subsec-discrete-EV-7">Having found the expected value for one named distribution, we may be tempted to run through our other discrete distributions (binomial, geometric, and Poisson) and find nice formulas for their expected values. Unfortunately, both the geometric distribution and the Poisson distribution have infinitely many possible values, and the summation of infinitely many discrete terms is beyond the scope of this course. (A Calculus 2 course covering series would give the right tools to find expected value formulas with full justification for these distributions.)<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="para" id="subsec-discrete-EV-8">The binomial distribution takes only finitely many values, and we have <a href="sec-Discrete-RVs.html#fact-binomial-probability-formula" class="xref" data-knowl="./knowl/xref/fact-binomial-probability-formula.html" data-reveal-label="Reveal" data-close-label="Close" title="Fact 2.1.12: Binomial Probability Formula">Fact 2.1.12</a> that lets us generate a distribution table reasonably quickly, given the parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> Feel free to pick some parameter values (keep <span class="process-math">\(n\)</span> fairly small!) to try this calculation by hand, but we’ll use a powerful theorem a bit later to give a very nice general formula depending on <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Subsection 3.1.1: Discrete Expected Value"><a href="#subsec-discrete-EV" title="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value">🔗</a></div></section><section class="subsection" id="subsec-continuous-EV"><h3 class="heading hide-type">
|
||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Continuous Expected Value</span>
|
||
</h3>
|
||
<div class="para" id="subsec-continuous-EV-2">Expected value for a discrete random variable is a weighted average of the values taken by the variable with weights provided by the probability distribution. We’d like to use the same idea for a continuous random variable, but now we’re averaging over an interval’s worth of values. Also, individual values don’t have probabilities, they have probability densities.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-continuous-EV"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="def-continuous-EV-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">\([a, b]\)</span> with pdf <span class="process-math">\(f(x)\text{.}\)</span> The <dfn class="terminology">expected value</dfn> of <span class="process-math">\(X\)</span> is given by:</div>
|
||
<div class="displaymath process-math" id="def-continuous-EV-1-1-6">
|
||
\begin{gather*}
|
||
\E(X) = \int_a^b x f(x)\ dx.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#def-continuous-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 3.1.6"><a href="#def-continuous-EV" title="Copy heading and permalink for Definition 3.1.6" aria-label="Copy heading and permalink for Definition 3.1.6">🔗</a></div></article><div class="para" id="subsec-continuous-EV-4">It’s worth viewing this definition side-by-side with <a href="sec-Expected-Value.html#def-discrete-EV" class="xref" data-knowl="./knowl/xref/def-discrete-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 3.1.1">Definition 3.1.1</a> to understand that they have analogous structure. The pdf values <span class="process-math">\(f(x)\)</span> are densities, not probabilities. But the product <span class="process-math">\(f(x)\ dx\)</span> is probability. So the expression <span class="process-math">\(x f(x)\ dx\)</span> is analogous to <span class="process-math">\(x_i \Pr(X = x_i)\text{:}\)</span> value times probability. Finally, the integral symbol <span class="process-math">\(\int\)</span> is meant to look like a stretched out letter "S" because it is a form of infinite summation. So, both <span class="process-math">\(\sum x_i \Pr(X = x_i)\)</span> and <span class="process-math">\(\int_a^b x f(x)\ dx\)</span> should be understood as taking the products of values with their probabilities and then summing up those products.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-EV-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-uniform-EV"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="example-uniform-EV-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> be uniform on the interval <span class="process-math">\([a, b]\text{.}\)</span> So <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span> We might reasonably expect that the average value of <span class="process-math">\(X\)</span> would be the midpoint of the interval. Let’s check that:</div>
|
||
<div class="displaymath process-math" id="example-uniform-EV-1-1-6">
|
||
\begin{align*}
|
||
\E(X) \amp = \int_a^b x f(x)\ dx \\
|
||
\amp = \int_a^b \frac{x}{b - a} \ dx \\
|
||
\amp = \frac{1}{b-a} \left(\frac{x^2}{2}\right)\bigg|_a^b \\
|
||
\amp = \frac{1}{b-a} \left( \frac{b^2}{2} - \frac{a^2}{2} \right) \\
|
||
\amp = \frac{1}{b - a}\left( \frac{(b - a)(b + a)}{2}\right) \\
|
||
\amp = \frac{a + b}{2}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-uniform-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.7"><a href="#example-uniform-EV" title="Copy heading and permalink for Example 3.1.7" aria-label="Copy heading and permalink for Example 3.1.7">🔗</a></div></article><article class="example example-like" id="subsec-continuous-EV-6"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.8</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="subsec-continuous-EV-6-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> take values in <span class="process-math">\([0, 4]\)</span> with pdf <span class="process-math">\(f(x) = \frac{3\sqrt{x}}{16}\text{.}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="subsec-continuous-EV-6-1-1-4">
|
||
\begin{align*}
|
||
\E(X) \amp = \int_0^4 x f(x)\ dx \\
|
||
\amp = \int_0^4 \frac{3x^{3/2}}{16}\ dx \\
|
||
\amp = \frac{3}{16}\left(\frac{x^{5/2}}{5/2}\right)\bigg|_0^4\\
|
||
\amp = \frac{3}{16}\left(\frac{2x^{5/2}}{5}\right)\bigg|_0^4\\
|
||
\amp = \frac{3}{40}\left(x^{5/2}\right)\bigg|_0^4\\
|
||
\amp = \frac{3}{40}(32 - 0) \\
|
||
\amp = \frac{12}{5} \\
|
||
\amp = 2.4.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-EV-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.8"><a href="#subsec-continuous-EV-6" title="Copy heading and permalink for Example 3.1.8" aria-label="Copy heading and permalink for Example 3.1.8">🔗</a></div></article><div class="autopermalink" data-description="Subsection 3.1.2: Continuous Expected Value"><a href="#subsec-continuous-EV" title="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value">🔗</a></div></section><section class="subsection" id="subsec-linearity-EV"><h3 class="heading hide-type">
|
||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Linearity of Expected Value</span>
|
||
</h3>
|
||
<div class="para logical" id="subsec-linearity-EV-2">
|
||
<div class="para">In the context of many areas of mathematics, the word "linearity" doesn’t refer to some graph being a straight line. Rather, it refers to a situation in which some mathematical object or operation behaves in the nicest possible way under the operations of addition and scalar multiplication (i.e., multiplication by a constant). You’ve already encountered linearity several times in Calculus 1, whether or not it was phrased that way.</div>
|
||
<ul class="disc" id="subsec-linearity-EV-2-1">
|
||
<li id="subsec-linearity-EV-2-1-1">
|
||
<div class="para logical" id="subsec-linearity-EV-2-1-1-1">
|
||
<div class="para">Limits are linear:</div>
|
||
<div class="displaymath process-math" id="subsec-linearity-EV-2-1-1-1-1">
|
||
\begin{align*}
|
||
\lim f(x) + g(x) \amp = \lim f(x) + \lim g(x)\\
|
||
\lim c f(x) \amp = c \lim f(x)
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item"><a href="#subsec-linearity-EV-2-1-1" title="Copy heading and permalink for Item" aria-label="Copy heading and permalink for Item">🔗</a></div>
|
||
</li>
|
||
<li id="subsec-linearity-EV-2-1-2">
|
||
<div class="para logical" id="subsec-linearity-EV-2-1-2-1">
|
||
<div class="para">Derivatives are linear:</div>
|
||
<div class="displaymath process-math" id="subsec-linearity-EV-2-1-2-1-1">
|
||
\begin{align*}
|
||
\frac{d}{dx}(f(x) + g(x)) \amp = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\\
|
||
\frac{d}{dx}(c f(x)) \amp = c \frac{d}{dx}(f(x))
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item"><a href="#subsec-linearity-EV-2-1-2" title="Copy heading and permalink for Item" aria-label="Copy heading and permalink for Item">🔗</a></div>
|
||
</li>
|
||
<li id="subsec-linearity-EV-2-1-3">
|
||
<div class="para logical" id="subsec-linearity-EV-2-1-3-1">
|
||
<div class="para">Integrals (both definite and indefinite) are linear:</div>
|
||
<div class="displaymath process-math" id="subsec-linearity-EV-2-1-3-1-1">
|
||
\begin{align*}
|
||
\int f(x) + g(x) \amp = \int f(x) + \int g(x)\\
|
||
\int c f(x) \amp = c \int f(x)
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2-1-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item"><a href="#subsec-linearity-EV-2-1-3" title="Copy heading and permalink for Item" aria-label="Copy heading and permalink for Item">🔗</a></div>
|
||
</li>
|
||
</ul>
|
||
<div class="para">Now, we see that expected value is also linear:</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="theorem theorem-like" id="thm-linearity-of-EV"><h4 class="heading">
|
||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.9</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">Linearity of Expectation.</span>
|
||
</h4>
|
||
<div class="para logical" id="thm-linearity-of-EV-2-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables and <span class="process-math">\(\alpha \in \R\text{.}\)</span> Then:</div>
|
||
<div class="displaymath process-math" id="thm-linearity-of-EV-2-1-4">
|
||
\begin{align*}
|
||
\E(X + Y) \amp = \E(X) + \E(Y) \\
|
||
\E(\alpha X) \amp = \alpha \E(X)
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-linearity-of-EV-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Theorem 3.1.9: Linearity of Expectation"><a href="#thm-linearity-of-EV" title="Copy heading and permalink for Theorem 3.1.9: Linearity of Expectation" aria-label="Copy heading and permalink for Theorem 3.1.9: Linearity of Expectation">🔗</a></div></article><div class="para" id="subsec-linearity-EV-4">Some application of the linearity of expectation will seem entirely reasonable:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-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-EV-sum-n-dice"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.10</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="example-EV-sum-n-dice-1-1">
|
||
<div class="para">In <a href="sec-Expected-Value.html#example-EV-die" class="xref" data-knowl="./knowl/xref/example-EV-die.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.2">Example 3.1.2</a>, we saw that the expected value of a fair 6-sided die roll is 3.5. In <a href="sec-Expected-Value.html#example-EV-sum-2-dice" class="xref" data-knowl="./knowl/xref/example-EV-sum-2-dice.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.3">Example 3.1.3</a>, we saw that the expected value of the sum of two die rolls is 7, which is precisely 2 times 3.5. The calculation in <a href="sec-Expected-Value.html#example-EV-sum-2-dice" class="xref" data-knowl="./knowl/xref/example-EV-sum-2-dice.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.3">Example 3.1.3</a> would be very tedious to recreate for three die rolls, let alone generalizing for <span class="process-math">\(n\)</span> rolls. However, suppose we define separate random variables <span class="process-math">\(R_1, R_2, \dotsc, R_n\)</span> for the result of each individual roll. Let <span class="process-math">\(S\)</span> be the sum of the <span class="process-math">\(n\)</span> rolls. Then <span class="process-math">\(S = R_1 + R_2 + \dotsb + R_n.\)</span> By linearity:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-EV-die.html ./knowl/xref/example-EV-sum-2-dice.html ./knowl/xref/example-EV-sum-2-dice.html" id="example-EV-sum-n-dice-1-1-9">
|
||
\begin{align*}
|
||
\E(S) \amp = \E(R_1 + R_2 + \dotsb + R_n) \\
|
||
\amp = \E(R_1) + \E(R_2) + \dotsb + \E(R_n) \\
|
||
\amp = \underbrace{3.5 + 3.5 + \dotsb + 3.5}_{n \text{ times}} \\
|
||
\amp = 3.5n.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">For two rolls, we recover our previous result: <span class="process-math">\(3.5(2) = 7.\)</span> But now we have a general formula that tells the expected sum for any number of rolls. It also gives a perfectly reasonable answer: the expected sum of <span class="process-math">\(n\)</span> rolls is <span class="process-math">\(n\)</span> times the expected value of a single roll.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-n-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.10"><a href="#example-EV-sum-n-dice" title="Copy heading and permalink for Example 3.1.10" aria-label="Copy heading and permalink for Example 3.1.10">🔗</a></div></article><article class="example example-like" id="example-binomial-EV"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="example-binomial-EV-1-1">
|
||
<div class="para">Suppose <span class="process-math">\(S \sim \Bin(n, p)\text{.}\)</span> <a href="sec-Discrete-RVs.html#def-binomial-distribution" class="xref" data-knowl="./knowl/xref/def-binomial-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.1.6">Definition 2.1.6</a> describes a binomially distributed random variable as counting the number of occurrences of some event, which either happens or not in each of <span class="process-math">\(n\)</span> independent trials. This leads us to a very natural idea: identify each individual possible occurrence of the event, and assign it an indicator random variable. In this case, let <span class="process-math">\(H_i\)</span> indicate that flip <span class="process-math">\(i\)</span> comes up heads. Then</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html" id="example-binomial-EV-1-1-6">
|
||
\begin{gather*}
|
||
S = H_1 + H_2 + \dotsb + H_n.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">We already know (<a href="sec-Expected-Value.html#example-indicator-EV" class="xref" data-knowl="./knowl/xref/example-indicator-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.5">Example 3.1.5</a>) that the expected value of each <span class="process-math">\(H_i\)</span> is the probability of the indicated event—that flip <span class="process-math">\(i\)</span> is heads. This is precisely the parameter <span class="process-math">\(p\text{.}\)</span> Therefore:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html" id="example-binomial-EV-1-1-12">
|
||
\begin{align*}
|
||
\E(S) \amp = \E(H_1 + H_2 + \dotsb + H_n) \\
|
||
\amp = \E(H_1) + \E(H_2) + \dotsb + \E(H_n) \\
|
||
\amp = \underbrace{p + p + \dotsb + p}_{n \text{ times}} \\
|
||
\amp = np.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Remember this formula! We’ll make frequent use of it.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-binomial-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.11"><a href="#example-binomial-EV" title="Copy heading and permalink for Example 3.1.11" aria-label="Copy heading and permalink for Example 3.1.11">🔗</a></div></article><div class="para" id="subsec-linearity-EV-7">The previous examples are so reasonable it may seem unimpressive. But consider the following use of linearity:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-7" 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-linearity-EV-8"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.12</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="subsec-linearity-EV-8-1-1">We say a flip sequence has a <dfn class="terminology">run of 4 heads starting at <span class="process-math">\(k\)</span></dfn> if flips <span class="process-math">\(k, k+1, k+2, k+3\)</span> are all heads. We don’t care about overlaps: the flip sequence <span class="process-math">\(HHHHH\)</span> has a run of 4 heads starting at 1 and another run starting at 2.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="subsec-linearity-EV-8-1-2">Suppose we flip a fair coin 100 times. How many runs of 4 heads should we expect?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-8-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions">
|
||
<details id="subsec-linearity-EV-8-2" class="hint solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Hint</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="hint solution-like knowl__content">
|
||
<div class="para" id="subsec-linearity-EV-8-2-1">Can you mimic the structure of <a href="sec-Expected-Value.html#example-binomial-EV" class="xref" data-knowl="./knowl/xref/example-binomial-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.11">Example 3.1.11</a>?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Hint 3.1.12.1"><a href="#subsec-linearity-EV-8-2" title="Copy heading and permalink for Hint 3.1.12.1" aria-label="Copy heading and permalink for Hint 3.1.12.1">🔗</a></div>
|
||
</div></details><details id="subsec-linearity-EV-8-3" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="subsec-linearity-EV-8-3-1">
|
||
<div class="para">Let <span class="process-math">\(R_k\)</span> indicate a run of 4 heads starting at <span class="process-math">\(k\text{.}\)</span> Since this requires 4 precise flips to come up heads, the probability of the indicated event should be</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-EV.html" id="subsec-linearity-EV-8-3-1-3">
|
||
\begin{gather*}
|
||
\left(\frac{1}{2}\right)^4 = \frac{1}{16}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">The very last starting flip for a run of 4 heads is flip number 97 (so that flips 97, 98, 99, and 100 would all be heads). Then <span class="process-math">\(S = R_1 + R_2 + \dotsb + R_{97}\)</span> precisely counts the number of runs of 4 heads. Using linearity:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-EV.html" id="subsec-linearity-EV-8-3-1-5">
|
||
\begin{align*}
|
||
\E(S) \amp = \E(R_1 + R_2 + \dotsb + R_{97}) \\
|
||
\amp = \E(R_1) + \E(R_2) + \dotsb + \E(R_{97}) \\
|
||
\amp = \underbrace{\frac{1}{16} + \frac{1}{16} + \dotsb + \frac{1}{16}}_{97 \text{ times}} \\
|
||
\amp = \frac{97}{16} \\
|
||
\amp \approx 6.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Notice that, unlike <a href="sec-Expected-Value.html#example-binomial-EV" class="xref" data-knowl="./knowl/xref/example-binomial-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.11">Example 3.1.11</a>, the events being indicated here are <em class="emphasis">not</em> independent from each other. Linearity is unaffected by the fact that that the runs of heads can overlap.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-8-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Solution 3.1.12.1"><a href="#subsec-linearity-EV-8-3" title="Copy heading and permalink for Solution 3.1.12.1" aria-label="Copy heading and permalink for Solution 3.1.12.1">🔗</a></div>
|
||
</div></details>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 3.1.12"><a href="#subsec-linearity-EV-8" title="Copy heading and permalink for Example 3.1.12" aria-label="Copy heading and permalink for Example 3.1.12">🔗</a></div></article><div class="autopermalink" data-description="Subsection 3.1.3: Linearity of Expected Value"><a href="#subsec-linearity-EV" title="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value">🔗</a></div></section><section class="exercises" id="exercises-Expected-Value"><h3 class="heading hide-type">
|
||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||
</h3>
|
||
<article class="exercise exercise-like" id="exercises-Expected-Value-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Expected-Value-1-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="exercises-Expected-Value-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.13<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.13: "><a href="#exercises-Expected-Value-1-1-2" title="Copy heading and permalink for Table 3.1.13: " aria-label="Copy heading and permalink for Table 3.1.13: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr class="header-horizontal">
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
</table></div></figure><div class="solutions"><details id="exercises-Expected-Value-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" id="exercises-Expected-Value-1-2-1">4.8.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.1.1"><a href="#exercises-Expected-Value-1-2" title="Copy heading and permalink for Answer 3.1.4.1.1" aria-label="Copy heading and permalink for Answer 3.1.4.1.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 3.1.4.1"><a href="#exercises-Expected-Value-1" title="Copy heading and permalink for Exercise 3.1.4.1" aria-label="Copy heading and permalink for Exercise 3.1.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Expected-Value-2-1-1">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables each taking the values 1, 2, 3, 4, 5. Write different distribution tables for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> so that they have the same expected value.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-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">
|
||
<figure class="table table-like" id="exercises-Expected-Value-2-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.14<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distributions<div class="autopermalink" data-description="Table 3.1.14: Example Distributions"><a href="#exercises-Expected-Value-2-2-1" title="Copy heading and permalink for Table 3.1.14: Example Distributions" aria-label="Copy heading and permalink for Table 3.1.14: Example Distributions">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(a\)</span></td>
|
||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = a)\)</span></td>
|
||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(Y = a)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="l m b0 r0 l0 t0 lines">1</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="l m b0 r0 l0 t0 lines">2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="l m b0 r0 l0 t0 lines">3</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.6</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="l m b0 r0 l0 t0 lines">4</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="l m b0 r0 l0 t0 lines">5</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
</table></div></figure><div class="autopermalink" data-description="Answer 3.1.4.2.1"><a href="#exercises-Expected-Value-2-2" title="Copy heading and permalink for Answer 3.1.4.2.1" aria-label="Copy heading and permalink for Answer 3.1.4.2.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 3.1.4.2"><a href="#exercises-Expected-Value-2" title="Copy heading and permalink for Exercise 3.1.4.2" aria-label="Copy heading and permalink for Exercise 3.1.4.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="introduction" id="exercises-Expected-Value-3-1"><div class="para" id="exercises-Expected-Value-3-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div></div>
|
||
<article class="task exercise-like" id="exercises-Expected-Value-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||
<div class="para" id="exercises-Expected-Value-3-2-1-1">If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\E(N)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-3-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Expected-Value-3-2-2-1">40.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.3.a.1"><a href="#exercises-Expected-Value-3-2-2" title="Copy heading and permalink for Answer 3.1.4.3.a.1" aria-label="Copy heading and permalink for Answer 3.1.4.3.a.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Task 3.1.4.3.a"><a href="#exercises-Expected-Value-3-2" title="Copy heading and permalink for Task 3.1.4.3.a" aria-label="Copy heading and permalink for Task 3.1.4.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Expected-Value-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||
<div class="para" id="exercises-Expected-Value-3-3-1-1">What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-3-3-2" 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-Expected-Value-3-3-2-1">48.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.3.b.1"><a href="#exercises-Expected-Value-3-3-2" title="Copy heading and permalink for Answer 3.1.4.3.b.1" aria-label="Copy heading and permalink for Answer 3.1.4.3.b.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Task 3.1.4.3.b"><a href="#exercises-Expected-Value-3-3" title="Copy heading and permalink for Task 3.1.4.3.b" aria-label="Copy heading and permalink for Task 3.1.4.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Expected-Value-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
|
||
<div class="para" id="exercises-Expected-Value-3-4-1-1">What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-3-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Expected-Value-3-4-2-1">100.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.3.c.1"><a href="#exercises-Expected-Value-3-4-2" title="Copy heading and permalink for Answer 3.1.4.3.c.1" aria-label="Copy heading and permalink for Answer 3.1.4.3.c.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Task 3.1.4.3.c"><a href="#exercises-Expected-Value-3-4" title="Copy heading and permalink for Task 3.1.4.3.c" aria-label="Copy heading and permalink for Task 3.1.4.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 3.1.4.3"><a href="#exercises-Expected-Value-3" title="Copy heading and permalink for Exercise 3.1.4.3" aria-label="Copy heading and permalink for Exercise 3.1.4.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Expected-Value-4-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\E(Y) = -2\text{,}\)</span> and <span class="process-math">\(\E(Z) = 1\text{,}\)</span> what is <span class="process-math">\(\E(4X + 5Y - Z + 3)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Expected-Value-4-2-1">4.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.4.1"><a href="#exercises-Expected-Value-4-2" title="Copy heading and permalink for Answer 3.1.4.4.1" aria-label="Copy heading and permalink for Answer 3.1.4.4.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 3.1.4.4"><a href="#exercises-Expected-Value-4" title="Copy heading and permalink for Exercise 3.1.4.4" aria-label="Copy heading and permalink for Exercise 3.1.4.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Expected-Value-5-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-5-2" 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-Expected-Value-5-2-1">2/3.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.5.1"><a href="#exercises-Expected-Value-5-2" title="Copy heading and permalink for Answer 3.1.4.5.1" aria-label="Copy heading and permalink for Answer 3.1.4.5.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 3.1.4.5"><a href="#exercises-Expected-Value-5" title="Copy heading and permalink for Exercise 3.1.4.5" aria-label="Copy heading and permalink for Exercise 3.1.4.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Expected-Value-6-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([-1, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{3x^2}{2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="exercises-Expected-Value-6-2" 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-Expected-Value-6-2-1">0.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-6-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 3.1.4.6.1"><a href="#exercises-Expected-Value-6-2" title="Copy heading and permalink for Answer 3.1.4.6.1" aria-label="Copy heading and permalink for Answer 3.1.4.6.1">🔗</a></div>
|
||
</div></details></div>
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<div class="autopermalink" data-description="Exercise 3.1.4.6"><a href="#exercises-Expected-Value-6" title="Copy heading and permalink for Exercise 3.1.4.6" aria-label="Copy heading and permalink for Exercise 3.1.4.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Expected-Value-7-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Expected-Value-7-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<div class="para" id="exercises-Expected-Value-7-2-1">
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<span class="process-math">\(\frac{4}{3}\ln(4) \approx 1.85.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-7-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 3.1.4.7.1"><a href="#exercises-Expected-Value-7-2" title="Copy heading and permalink for Answer 3.1.4.7.1" aria-label="Copy heading and permalink for Answer 3.1.4.7.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 3.1.4.7"><a href="#exercises-Expected-Value-7" title="Copy heading and permalink for Exercise 3.1.4.7" aria-label="Copy heading and permalink for Exercise 3.1.4.7">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Expected-Value-8-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> In a previous problem, you found the value of <span class="process-math">\(k\text{.}\)</span> Now, find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Expected-Value-8-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<div class="para" id="exercises-Expected-Value-8-2-1">
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<span class="process-math">\(E(X) \approx 3.04.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Answer 3.1.4.8.1"><a href="#exercises-Expected-Value-8-2" title="Copy heading and permalink for Answer 3.1.4.8.1" aria-label="Copy heading and permalink for Answer 3.1.4.8.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 3.1.4.8"><a href="#exercises-Expected-Value-8" title="Copy heading and permalink for Exercise 3.1.4.8" aria-label="Copy heading and permalink for Exercise 3.1.4.8">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-9"><h4 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Expected-Value-9-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Expected-Value-9-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<div class="para" id="exercises-Expected-Value-9-2-1">
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<span class="process-math">\(\frac{1}{2}\left( \ln(2) + \frac{7}{3}\right) \approx 1.513.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-9-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Answer 3.1.4.9.1"><a href="#exercises-Expected-Value-9-2" title="Copy heading and permalink for Answer 3.1.4.9.1" aria-label="Copy heading and permalink for Answer 3.1.4.9.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 3.1.4.9"><a href="#exercises-Expected-Value-9" title="Copy heading and permalink for Exercise 3.1.4.9" aria-label="Copy heading and permalink for Exercise 3.1.4.9">🔗</a></div></article><div class="autopermalink" data-description="Exercises 3.1.4"><a href="#exercises-Expected-Value" title="Copy heading and permalink for Exercises 3.1.4" aria-label="Copy heading and permalink for Exercises 3.1.4">🔗</a></div></section><div class="autopermalink" data-description="Section 3.1: Expected Value"><a href="#sec-Expected-Value" title="Copy heading and permalink for Section 3.1: Expected Value" aria-label="Copy heading and permalink for Section 3.1: Expected Value">🔗</a></div></section></div>
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