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@@ -366,9 +366,64 @@ eBookConfig.enable_chatcodes = false;
<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-Joint-Distributions-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-Joint-Distributions-1-1-1">TODO<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="exercises-Joint-Distributions-1-1-1">Consider <span class="process-math">\(X, Y\)</span> with the joint distribution table below. Are <span class="process-math">\(X, Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-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-Joint-Distributions-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint distribution for <span class="process-math">\(X, Y\)</span><div class="autopermalink" data-description="Table 2.3.6: Joint distribution for X, Y"><a href="#exercises-Joint-Distributions-1-1-2" title="Copy heading and permalink for Table 2.3.6: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 2.3.6: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.2</td>
<td class="c m b1 r0 l0 t0 lines">0.3</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.4</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="solutions"><details id="exercises-Joint-Distributions-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-Joint-Distributions-1-2-1">No. For example, <span class="process-math">\(\Pr(X = 0, Y = 0) \neq \Pr(X = 0)\Pr(Y = 0).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-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="Exercise 2.3.1"><a href="#exercises-Joint-Distributions-1" title="Copy heading and permalink for Exercise 2.3.1" aria-label="Copy heading and permalink for Exercise 2.3.1">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.3"><a href="#exercises-Joint-Distributions" title="Copy heading and permalink for Exercises 2.3" aria-label="Copy heading and permalink for Exercises 2.3">🔗</a></div></section><div class="autopermalink" data-description="Section 2.3: Joint Distributions"><a href="#sec-Joint-Distributions" title="Copy heading and permalink for Section 2.3: Joint Distributions" aria-label="Copy heading and permalink for Section 2.3: Joint Distributions">🔗</a></div></section></div>
<div class="autopermalink" data-description="Answer 2.3.1.1"><a href="#exercises-Joint-Distributions-1-2" title="Copy heading and permalink for Answer 2.3.1.1" aria-label="Copy heading and permalink for Answer 2.3.1.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.3.1"><a href="#exercises-Joint-Distributions-1" title="Copy heading and permalink for Exercise 2.3.1" aria-label="Copy heading and permalink for Exercise 2.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Joint-Distributions-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para logical" id="exercises-Joint-Distributions-2-1-1">
<div class="para">Suppose <span class="process-math">\(X, Y\)</span> have the distributions:</div>
<div class="displaymath process-math" id="exercises-Joint-Distributions-2-1-1-2">
\begin{align*}
\Pr(X = 0) \amp = 0.1 \amp \Pr(Y = 0) \amp = 0.4 \\
\Pr(X = 1) \amp = 0.4 \amp \Pr(Y = 1) \amp = 0.6 \\
\Pr(X = 2) \amp = 0.5
\end{align*}
</div>
<div class="para">Assuming <span class="process-math">\(X, Y\)</span> are independent, write a joint distribution table.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-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-Joint-Distributions-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-Joint-Distributions-2-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 2.3.7: Joint Distribution"><a href="#exercises-Joint-Distributions-2-2-1" title="Copy heading and permalink for Table 2.3.7: Joint Distribution" aria-label="Copy heading and permalink for Table 2.3.7: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.04</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.06</td>
<td class="c m b0 r1 l0 t0 lines">0.24</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 2.3.2.1"><a href="#exercises-Joint-Distributions-2-2" title="Copy heading and permalink for Answer 2.3.2.1" aria-label="Copy heading and permalink for Answer 2.3.2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.3.2"><a href="#exercises-Joint-Distributions-2" title="Copy heading and permalink for Exercise 2.3.2" aria-label="Copy heading and permalink for Exercise 2.3.2">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.3"><a href="#exercises-Joint-Distributions" title="Copy heading and permalink for Exercises 2.3" aria-label="Copy heading and permalink for Exercises 2.3">🔗</a></div></section><div class="autopermalink" data-description="Section 2.3: Joint Distributions"><a href="#sec-Joint-Distributions" title="Copy heading and permalink for Section 2.3: Joint Distributions" aria-label="Copy heading and permalink for Section 2.3: Joint Distributions">🔗</a></div></section></div>
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