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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Expected-Value"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Expected Value</span>
</h2>
<section class="subsection" id="subsec-discrete-EV"><h3 class="heading hide-type">
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Discrete Expected Value</span>
</h3>
<div class="para" id="subsec-discrete-EV-2">A full probability distribution (for a discrete random variable) or a density function (for a continuous random variable) carry all of the probability information. Often, we seek <dfn class="terminology">statistics</dfn>, i.e., numbers that summarize a distribution in some way.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="definition definition-like" id="def-discrete-expected-value"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="def-discrete-expected-value-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> (also called <dfn class="terminology">mean</dfn>, or <dfn class="terminology">expectation</dfn>) of <span class="process-math">\(X\)</span> is the weighted average of the values of <span class="process-math">\(X\text{,}\)</span> where the weights are the probabilities of <span class="process-math">\(X\)</span> taking each value:</div>
<div class="displaymath process-math" id="def-discrete-expected-value-1-1-9">
\begin{gather*}
\E(X) = \sum_{i=1}^n x_i \Pr(X = x_i).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-expected-value-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 3.1.1"><a href="#def-discrete-expected-value" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div></article><article class="example example-like" id="example-EV-die"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="example-EV-die-1-1">
<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
<div class="displaymath process-math" id="example-EV-die-1-1-4">
\begin{align*}
\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
\end{align*}
</div>
<div class="para">For example, the expected value of a fair 6-sided die roll <span class="process-math">\(R\)</span> is:</div>
<div class="displaymath process-math" id="example-EV-die-1-1-6">
\begin{gather*}
\E(R) = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><article class="example example-like" id="example-EV-sum-2-dice"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="example-EV-sum-2-dice-1-1">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="example-EV-sum-2-dice-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k \cdot \Pr(S = k)\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(20/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(42/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(40/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(36/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">10</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">11</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(22/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">12</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
</tr>
</table></div></figure> <div class="para" id="example-EV-sum-2-dice-1-3">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><article class="example example-like" id="example-indicator-EV"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="example-indicator-EV-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="example-indicator-EV-1-1-4">
\begin{gather*}
\E(X) = 0 \cdot (1-p) + 1 \cdot p = p.
\end{gather*}
</div>
<div class="para">That is, the expected value of an indicator random variable is the probability of the event that it indicates.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 3.1.5"><a href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div></article><div class="para" id="subsec-discrete-EV-7">Having found the expected value for one named distribution, we may be tempted to run through our other discrete distributions (binomial, geometric, and Poisson) and find nice formulas for their expected values. Unfortunately, both the geometric distribution and the Poisson distribution have infinitely many possible values, and the summation of infinitely many discrete terms is beyond the scope of this course. (A Calculus 2 course covering series would give the right tools to find expected value formulas with full justification for these distributions.)<div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="subsec-discrete-EV-8">The binomial distribution takes only finitely many values, and we have <a href="sec-Discrete-RVs.html#fact-binomial-probability-formula" class="xref" data-knowl="./knowl/xref/fact-binomial-probability-formula.html" data-reveal-label="Reveal" data-close-label="Close" title="Fact 2.1.12: Binomial Probability Formula">Fact 2.1.12</a> that lets us generate a distribution table reasonably quickly, given the parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> Feel free to pick some parameter values (keep <span class="process-math">\(n\)</span> fairly small!) to try this calculation by hand, but we’ll use a powerful theorem a bit later to give a very nice general formula depending on <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-discrete-EV-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Subsection 3.1.1: Discrete Expected Value"><a href="#subsec-discrete-EV" title="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.1: Discrete Expected Value">🔗</a></div></section><section class="subsection" id="subsec-continuous-EV"><h3 class="heading hide-type">
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Continuous Expected Value</span>
</h3>
<div class="para" id="subsec-continuous-EV-2"><div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div>
<div class="autopermalink" data-description="Subsection 3.1.2: Continuous Expected Value"><a href="#subsec-continuous-EV" title="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.2: Continuous Expected Value">🔗</a></div></section><section class="subsection" id="subsec-linearity-EV"><h3 class="heading hide-type">
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Linearity of Expected Value</span>
</h3>
<div class="para" id="subsec-linearity-EV-2"><div class="autopermalink" data-description="Paragraph"><a href="#subsec-linearity-EV-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div></div>
<div class="autopermalink" data-description="Subsection 3.1.3: Linearity of Expected Value"><a href="#subsec-linearity-EV" title="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value" aria-label="Copy heading and permalink for Subsection 3.1.3: Linearity of Expected Value">🔗</a></div></section><section class="exercises" id="exercises-Expected-Value"><h3 class="heading hide-type">
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercises-Expected-Value-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-1-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="exercises-Expected-Value-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.6: "><a href="#exercises-Expected-Value-1-1-2" title="Copy heading and permalink for Table 3.1.6: " aria-label="Copy heading and permalink for Table 3.1.6: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 3.1.4.1"><a href="#exercises-Expected-Value-1" title="Copy heading and permalink for Exercise 3.1.4.1" aria-label="Copy heading and permalink for Exercise 3.1.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-2-1-1">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables each taking the values 1, 2, 3, 4, 5. Write different distribution tables for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> so that they have the same expected value.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.2"><a href="#exercises-Expected-Value-2" title="Copy heading and permalink for Exercise 3.1.4.2" aria-label="Copy heading and permalink for Exercise 3.1.4.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-3-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads. If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\E(N)\text{?}\)</span> What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span> What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.3"><a href="#exercises-Expected-Value-3" title="Copy heading and permalink for Exercise 3.1.4.3" aria-label="Copy heading and permalink for Exercise 3.1.4.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-4-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\E(Y) = -2\text{,}\)</span> and <span class="process-math">\(\E(Z) = 1\text{,}\)</span> what is <span class="process-math">\(\E(4X + 5Y - Z + 3)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.4"><a href="#exercises-Expected-Value-4" title="Copy heading and permalink for Exercise 3.1.4.4" aria-label="Copy heading and permalink for Exercise 3.1.4.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-5-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.5"><a href="#exercises-Expected-Value-5" title="Copy heading and permalink for Exercise 3.1.4.5" aria-label="Copy heading and permalink for Exercise 3.1.4.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-6-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([-1, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{3x^2}{2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.6"><a href="#exercises-Expected-Value-6" title="Copy heading and permalink for Exercise 3.1.4.6" aria-label="Copy heading and permalink for Exercise 3.1.4.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-7-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.7"><a href="#exercises-Expected-Value-7" title="Copy heading and permalink for Exercise 3.1.4.7" aria-label="Copy heading and permalink for Exercise 3.1.4.7">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-8-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> In a previous problem, you found the value of <span class="process-math">\(k\text{.}\)</span> Now, find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 3.1.4.8"><a href="#exercises-Expected-Value-8" title="Copy heading and permalink for Exercise 3.1.4.8" aria-label="Copy heading and permalink for Exercise 3.1.4.8">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Expected-Value-9"><h4 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Expected-Value-9-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Expected-Value-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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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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