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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-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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<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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<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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<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 active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<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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<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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<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>
|
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<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>
|
||
<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-appendix">
|
||
<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
|
||
</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-Variance"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Variance</span>
|
||
</h2>
|
||
<div class="para" id="sec-Variance-2">The expected value is one of several statistics sometimes referred to as a "measure of central tendency". These are statistics that try to capture the "typical" value of the random variable. Our next statistic tries to measure how spread out a distribution is.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-variance"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="def-variance-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. The <dfn class="terminology">variance</dfn> of <span class="process-math">\(X\)</span> is:</div>
|
||
<div class="displaymath process-math" id="def-variance-1-1-4">
|
||
\begin{gather*}
|
||
\Var(X) = \E\left[ \left(X - \E(X)\right)^2 \right].
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#def-variance-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 3.2.1"><a href="#def-variance" title="Copy heading and permalink for Definition 3.2.1" aria-label="Copy heading and permalink for Definition 3.2.1">🔗</a></div></article><div class="para" id="sec-Variance-4">This formula tries to capture some measure of how spread out the distribution is. One way we might hope to measure spread is to calculate how far away <span class="process-math">\(X\)</span> is from its expected value on average. Squaring the term <span class="process-math">\(X - \E(X)\)</span> ensures that any difference between the value of the random variable and its expected value contributes positively. ("Distance away from" should be positive.) So, the formula in <a href="sec-Variance.html#def-variance" class="xref" data-knowl="./knowl/xref/def-variance.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 3.2.1">Definition 3.2.1</a> is conceptually clear. In practice, there’s another way to structure the formula that makes for cleaner calculation:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-4" 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-variance-formula"><h3 class="heading">
|
||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="thm-variance-formula-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. Then:</div>
|
||
<div class="displaymath process-math" id="thm-variance-formula-1-1-2">
|
||
\begin{gather*}
|
||
\Var(X) = \E\left(X^2\right) - \left(\E(X)\right)^2.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-formula-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="Theorem 3.2.2"><a href="#thm-variance-formula" title="Copy heading and permalink for Theorem 3.2.2" aria-label="Copy heading and permalink for Theorem 3.2.2">🔗</a></div></article><details id="sec-Variance-6" class="hiddenproof born-hidden-knowl"><summary class="knowl__link"><h3 class="heading"><span class="type">Proof<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3></summary><article class="hiddenproof knowl__content"><div class="para logical" id="sec-Variance-6-1">
|
||
<div class="displaymath process-math" id="sec-Variance-6-1-1">
|
||
\begin{align*}
|
||
\Var(X) \amp = \E\left[ \left(X - \E(X)\right)^2 \right] \\
|
||
\amp = \E\left[ X^2 - 2 \E(X) X + \left(\E(X)\right)^2\right] \\
|
||
\amp = \E\left(X^2\right) - 2 \E(X) \E(X) + \left(\E(X)\right)^2\\
|
||
\amp = \E\left(X^2\right) - \left(\E(X)\right)^2.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-6-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Proof 3.2.1"><a href="#sec-Variance-6" title="Copy heading and permalink for Proof 3.2.1" aria-label="Copy heading and permalink for Proof 3.2.1">🔗</a></div></article></details><article class="example example-like" id="example-var-die"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-var-die-1-1">
|
||
<div class="para">Let <span class="process-math">\(R\)</span> be the value of a fair 6-sided die roll. We’ve seen 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> that <span class="process-math">\(\E(R) = \frac{7}{2}\text{.}\)</span> To find <span class="process-math">\(\E\left(R^2\right)\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-EV-die.html" id="example-var-die-1-1-5">
|
||
\begin{align*}
|
||
\E\left(R^2\right) \amp = 1^2 \left(\frac{1}{6}\right) + 2^2 \left(\frac{1}{6}\right) + 3^2 \left(\frac{1}{6}\right) + 4^2 \left(\frac{1}{6}\right) + 5^2 \left(\frac{1}{6}\right) + 6^2 \left(\frac{1}{6}\right) \\
|
||
\amp = 1 \left(\frac{1}{6}\right) + 4 \left(\frac{1}{6}\right) + 9 \left(\frac{1}{6}\right) + 16 \left(\frac{1}{6}\right) + 25 \left(\frac{1}{6}\right) + 36 \left(\frac{1}{6}\right) \\
|
||
\amp = \frac{91}{6}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Then</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-EV-die.html" id="example-var-die-1-1-6">
|
||
\begin{gather*}
|
||
\Var(R) = \E\left(R^2\right) - \left(\E(R)\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12}.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-var-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 3.2.3"><a href="#example-var-die" title="Copy heading and permalink for Example 3.2.3" aria-label="Copy heading and permalink for Example 3.2.3">🔗</a></div></article><div class="para" id="sec-Variance-8">Notice that in the <span class="process-math">\(\E\left(R^2\right)\)</span> calculation, we’re just taking the basic structure of the <span class="process-math">\(\E(R)\)</span> calculation and squaring the <span class="process-math">\(R\)</span> values. The analogous calculation for a continuous random variable looks similar:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-8" 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-variance"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-uniform-variance-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> and we saw in <a href="sec-Expected-Value.html#example-uniform-EV" class="xref" data-knowl="./knowl/xref/example-uniform-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.7">Example 3.1.7</a> that <span class="process-math">\(\E(X) = \frac{a + b}{2}\text{.}\)</span> Now:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-uniform-EV.html" id="example-uniform-variance-1-1-7">
|
||
\begin{align*}
|
||
\E\left(X^2\right) \amp = \int_a^b x^2 f(x)\ dx \\
|
||
\amp = \int_a^b \frac{x^2}{b - a} \ dx \\
|
||
\amp = \frac{1}{b-a} \left(\frac{x^3}{3}\right)\bigg|_a^b \\
|
||
\amp = \frac{1}{b-a} \left( \frac{b^3}{3} - \frac{a^3}{3} \right) \\
|
||
\amp = \frac{1}{b - a}\left( \frac{(b - a)(a^2 + ab + b^2)}{3}\right) \\
|
||
\amp = \frac{a^2 + ab + b^2}{3}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Then:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-uniform-EV.html" id="example-uniform-variance-1-1-8">
|
||
\begin{align*}
|
||
\Var(X) \amp = \E\left(X^2\right) - \left(\E(X)\right)^2 \\
|
||
\amp = \frac{a^2 + ab + b^2}{3} - \frac{(a+b)^2}{4} \\
|
||
\amp = \frac{a^2 + ab + b^2}{3} - \frac{a^2 + 2ab + b^2}{4} \\
|
||
\amp = \frac{4a^2 + 4ab + 4b^2}{12} - \frac{3a^2 + 6ab + 3b^2}{12} \\
|
||
\amp = \frac{a^2 - 2ab + b^2}{12} \\
|
||
\amp = \frac{(a - b)^2}{12}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-uniform-variance-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 3.2.4"><a href="#example-uniform-variance" title="Copy heading and permalink for Example 3.2.4" aria-label="Copy heading and permalink for Example 3.2.4">🔗</a></div></article><div class="para" id="sec-Variance-10">Variance is <em class="emphasis">not</em> linear. In general, you can’t split up variance across plus or minus signs, and you can’t pull multiplied constants out of the variance like you can with expected value. However, there are some properties we can use to do similar manipulations to the variance.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-10" 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-variance-properties"><h3 class="heading">
|
||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="thm-variance-properties-1-1">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables, and let <span class="process-math">\(\alpha \in \R\text{.}\)</span> Then:</div>
|
||
<ol class="decimal" id="thm-variance-properties-1-1-4">
|
||
<li id="thm-variance-properties-1-1-4-1">
|
||
<div class="para" id="thm-variance-properties-1-1-4-1-1">
|
||
<span class="process-math">\(\Var(X + \alpha) = \Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-properties-1-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item 1"><a href="#thm-variance-properties-1-1-4-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||
</li>
|
||
<li id="thm-variance-properties-1-1-4-2">
|
||
<div class="para" id="thm-variance-properties-1-1-4-2-1">
|
||
<span class="process-math">\(\Var(\alpha X) = \alpha^2 \Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-properties-1-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item 2"><a href="#thm-variance-properties-1-1-4-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||
</li>
|
||
<li id="thm-variance-properties-1-1-4-3">
|
||
<div class="para" id="thm-variance-properties-1-1-4-3-1">If <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are independent, then <span class="process-math">\(\Var(X + Y) = \Var(X) + \Var(Y)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-properties-1-1-4-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item 3"><a href="#thm-variance-properties-1-1-4-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
|
||
</li>
|
||
</ol>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-properties-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="Theorem 3.2.5"><a href="#thm-variance-properties" title="Copy heading and permalink for Theorem 3.2.5" aria-label="Copy heading and permalink for Theorem 3.2.5">🔗</a></div></article><div class="para" id="sec-Variance-12">We’ll omit the proofs and just casually observe that these are reasonable properties. <span class="process-math">\(X + \alpha\)</span> is a shift of <span class="process-math">\(X\text{,}\)</span> and shifting shouldn’t change how spread out the distribution is. <span class="process-math">\(\alpha X\)</span> scaled <span class="process-math">\(X\)</span> by a constant, so it should scale the spread of the distribution by the same constant. But recall that the variance calculation involves squaring; it’s an indirect measure of spread. So it’s reasonable that scaling the distribution by a constant should scale the variance by its square.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-12" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="para" id="sec-Variance-13">Let’s put these properties to use in a pair of related examples:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Variance-13" 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-indicator-variance"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-indicator-variance-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> In <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>, we saw <span class="process-math">\(\E(X) = p\text{.}\)</span> Notice that <span class="process-math">\(X\)</span> takes the values 0 and 1 (with probabilities <span class="process-math">\(1 - p\)</span> and <span class="process-math">\(p\text{,}\)</span> respectively). But <span class="process-math">\(0^2 = 0\text{,}\)</span> and <span class="process-math">\(1^1 = 1\text{,}\)</span> so the expected value calculation for <span class="process-math">\(X^2\)</span> is precisely the same as for <span class="process-math">\(X\text{!}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html" id="example-indicator-variance-1-1-13">
|
||
\begin{align*}
|
||
\E\left(X^2\right) \amp = 0^2 \cdot (1-p) + 1^2 \cdot p = p.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">So:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html" id="example-indicator-variance-1-1-14">
|
||
\begin{align*}
|
||
\Var(X) \amp \E\left(X^2\right) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p).
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-variance-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 3.2.6"><a href="#example-indicator-variance" title="Copy heading and permalink for Example 3.2.6" aria-label="Copy heading and permalink for Example 3.2.6">🔗</a></div></article><article class="example example-like" id="example-binomial-variance"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-binomial-variance-1-1">
|
||
<div class="para">Recall the setup from <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>: a binomial random variable <span class="process-math">\(S\)</span> is the sum <span class="process-math">\(H_1 + H_2 + \dotsb + H_n\)</span> of <span class="process-math">\(n\)</span> indicator random variables, each with parameter <span class="process-math">\(p\text{.}\)</span> The results of different coin flips are independent from each other, so the random variables <span class="process-math">\(H_1, \dotsc, H_n\)</span> are independent from each other. From <a href="sec-Variance.html#example-indicator-variance" class="xref" data-knowl="./knowl/xref/example-indicator-variance.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.2.6">Example 3.2.6</a>, we know that <span class="process-math">\(\Var(H_i) = p(1-p)\)</span> for each <span class="process-math">\(i\text{.}\)</span> Therefore:</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-EV.html ./knowl/xref/example-indicator-variance.html" id="example-binomial-variance-1-1-10">
|
||
\begin{align*}
|
||
\Var(S) \amp = \Var(H_1 + H_2 + \dotsb + H_n) \\
|
||
\amp = \Var(H_1) + \Var(H_2) + \dotsb + \Var(H_n) \\
|
||
\amp = \underbrace{p(1-p) + p(1-p) + \dotsb + p(1-p)}_{n \text{ times}} \\
|
||
\amp = np(1-p).
|
||
\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-variance-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 3.2.7"><a href="#example-binomial-variance" title="Copy heading and permalink for Example 3.2.7" aria-label="Copy heading and permalink for Example 3.2.7">🔗</a></div></article><section class="exercises" id="exercises-Variance"><h3 class="heading hide-type">
|
||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||
</h3>
|
||
<article class="exercise exercise-like" id="exercises-Variance-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-1-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="exercises-Variance-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.2.8: "><a href="#exercises-Variance-1-1-2" title="Copy heading and permalink for Table 3.2.8: " aria-label="Copy heading and permalink for Table 3.2.8: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr class="header-horizontal">
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.15</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">6</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">9</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
</table></div></figure><div class="autopermalink" data-description="Exercise 3.2.1"><a href="#exercises-Variance-1" title="Copy heading and permalink for Exercise 3.2.1" aria-label="Copy heading and permalink for Exercise 3.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-2-1-1">Let <span class="process-math">\(X\)</span> be a random variable taking the values 1, 2, 3, 4, 5. Write a distribution table for <span class="process-math">\(X\text{,}\)</span> then use your table to write a distribution for <span class="process-math">\(X^2\text{.}\)</span> Then, find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.2"><a href="#exercises-Variance-2" title="Copy heading and permalink for Exercise 3.2.2" aria-label="Copy heading and permalink for Exercise 3.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-3-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads. If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\Var(N)\text{?}\)</span> What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span> What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.3"><a href="#exercises-Variance-3" title="Copy heading and permalink for Exercise 3.2.3" aria-label="Copy heading and permalink for Exercise 3.2.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-4-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\Var(X) = 2\text{,}\)</span> what is <span class="process-math">\(\E(X^2)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.4"><a href="#exercises-Variance-4" title="Copy heading and permalink for Exercise 3.2.4" aria-label="Copy heading and permalink for Exercise 3.2.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-5-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.5"><a href="#exercises-Variance-5" title="Copy heading and permalink for Exercise 3.2.5" aria-label="Copy heading and permalink for Exercise 3.2.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-6-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([-1, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{3x^2}{2}\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.6"><a href="#exercises-Variance-6" title="Copy heading and permalink for Exercise 3.2.6" aria-label="Copy heading and permalink for Exercise 3.2.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-7-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.7"><a href="#exercises-Variance-7" title="Copy heading and permalink for Exercise 3.2.7" aria-label="Copy heading and permalink for Exercise 3.2.7">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-8-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> In a previous problem, you found the value of <span class="process-math">\(k\text{.}\)</span> Now, find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.8"><a href="#exercises-Variance-8" title="Copy heading and permalink for Exercise 3.2.8" aria-label="Copy heading and permalink for Exercise 3.2.8">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Variance-9"><h4 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Variance-9-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Variance-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 3.2.9"><a href="#exercises-Variance-9" title="Copy heading and permalink for Exercise 3.2.9" aria-label="Copy heading and permalink for Exercise 3.2.9">🔗</a></div></article><div class="autopermalink" data-description="Exercises 3.2"><a href="#exercises-Variance" title="Copy heading and permalink for Exercises 3.2" aria-label="Copy heading and permalink for Exercises 3.2">🔗</a></div></section><div class="autopermalink" data-description="Section 3.2: Variance"><a href="#sec-Variance" title="Copy heading and permalink for Section 3.2: Variance" aria-label="Copy heading and permalink for Section 3.2: Variance">🔗</a></div></section></div>
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