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@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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@@ -654,30 +654,126 @@ S = H_1 + H_2 + \dotsb + H_n.
<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>
</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="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 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="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="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>
<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="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">
<div class="para" id="exercises-Expected-Value-7-2-1">
<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>
</div>
<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>
</div></details></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="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">
<div class="para" id="exercises-Expected-Value-8-2-1">
<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 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="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">
<div class="para" id="exercises-Expected-Value-9-2-1">
<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 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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