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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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</div></section><section class="onepage" id="quiz-05-3"><article class="exercise exercise-like" id="quiz-05-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="quiz-05-3-1-1-1">The joint and marginal distributions of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are given below. Find <span class="process-math">\(\Cov(X, Y)\)</span> and <span class="process-math">\(\rho_{X, Y}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="quiz-05-3-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">127<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 Table<div class="autopermalink" data-description="Table 127: Joint Distribution Table"><a href="#quiz-05-3-1-1-2" title="Copy heading and permalink for Table 127: Joint Distribution Table" aria-label="Copy heading and permalink for Table 127: Joint Distribution Table">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(Y = 2\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(Y = 3\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
</tr>
</table></div></figure> <figure class="table table-like" id="quiz-05-3-1-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">128<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution Table for <span class="process-math">\(X\)</span><div class="autopermalink" data-description="Table 128: Distribution Table for X"><a href="#quiz-05-3-1-1-3" title="Copy heading and permalink for Table 128: Distribution Table for X" aria-label="Copy heading and permalink for Table 128: Distribution Table for X">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 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">\(0\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.35\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.65\)</span></td>
</tr>
</table></div></figure> <figure class="table table-like" id="quiz-05-3-1-1-4"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">129<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution Table for <span class="process-math">\(Y\)</span><div class="autopermalink" data-description="Table 129: Distribution Table for Y"><a href="#quiz-05-3-1-1-4" title="Copy heading and permalink for Table 129: Distribution Table for Y" aria-label="Copy heading and permalink for Table 129: Distribution Table for Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(y\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(Y = y)\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.35\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.4\)</span></td>
</tr>
</table></div></figure><div class="solutions"><details id="quiz-05-3-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-05-3-1-2-1">
<div class="para">
<span class="process-math">\(\Cov(X, Y) = \E(XY) - \E(X)\E(Y)\text{.}\)</span> We have:</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-2">
\begin{align*}
\E(X) \amp = 0(0.35) + 1(0.65) = 0.65 \\
\E(Y) \amp = 1(0.25) + 2(0.35) + 3(0.4) = 2.15 \\
\E(XY) \amp = (0)(1)(0.1) + (0)(2)(0.15) + (0)(3)(0.1) + \\
\amp \quad\, (1)(1)(0.15) + (1)(2)(0.2) + (1)(3)(0.3) = 1.45 \\ \\
\Cov(X, Y) \amp = 1.45 - (0.65)(2.15) = \boxed{0.0525}
\end{align*}
</div>
<div class="para">Since <span class="process-math">\(X\)</span> is an indicator random variable, we have <span class="process-math">\(\Var(X) = (0.65)(0.35) = 0.2275\text{.}\)</span> For <span class="process-math">\(\Var(Y)\text{:}\)</span>
</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-6">
\begin{align*}
\E(Y^2) \amp = 1^2(0.25) + 2^2(0.35) + 3^2(0.4) = 5.25 \\
\Var(Y) \amp = \E(Y^2) - (\E(Y))^2 = 5.25 - (2.15)^2 = 0.6275
\end{align*}
</div>
<div class="para">So the correlation is:</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-7">
\begin{align*}
\rho_{X, Y} \amp = \frac{\Cov(X, Y)}{\sqrt{\Var(X)\Var(Y)}} = \frac{0.0525}{\sqrt{(0.2275)(0.6275)}} \approx \boxed{0.139}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-3-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 1.1"><a href="#quiz-05-3-1-2" title="Copy heading and permalink for Solution 1.1" aria-label="Copy heading and permalink for Solution 1.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-05-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><section class="onepage" id="quiz-05-4"><article class="exercise exercise-like" id="quiz-05-4-1"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="quiz-05-4-1-1-1">A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="quiz-05-4-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">130<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Sample Data<div class="autopermalink" data-description="Table 130: Sample Data"><a href="#quiz-05-4-1-1-2" title="Copy heading and permalink for Table 130: Sample Data" aria-label="Copy heading and permalink for Table 130: Sample Data">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\(x_i\)</span></th>
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\(y_i\)</span></th>
<th scope="col" class="l m b1 r0 l0 t0 lines">best fit predicted <span class="process-math">\(y_i\)</span>
</th>
<th scope="col" class="l m b1 r0 l0 t0 lines">res<span class="process-math">\(^2\)</span>
</th>
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\((y - \textrm{avg }y)^2\)</span></th>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(1.94\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(3.31\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(10.53\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(52.13\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(886.55\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(2.63\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(10.38\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(17.37\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(48.76\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(515.34\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(3.94\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(19.47\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(30.31\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(117.35\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(185.24 \)</span></td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(6.35\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(53.42\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(54.22\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(0.64\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(413.51\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">sum:</td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(992.55\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(992.55\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(953.03\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(38397.35\)</span></td>
</tr>
</table></div></figure><div class="solutions"><details id="quiz-05-4-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-05-4-1-2-1">
<div class="para">The RSS is the sum of res<span class="process-math">\(^2\text{,}\)</span> <span class="process-math">\(953.03\text{,}\)</span> and the SST is the sum of <span class="process-math">\((y - \textrm{avg})^2\text{,}\)</span> <span class="process-math">\(38397.35\text{.}\)</span> Then the coefficient of determination is:</div>
<div class="displaymath process-math" id="quiz-05-4-1-2-1-5">
\begin{gather*}
r^2 = 1 - \frac{\text{RSS}}{\text{SST}} = 1 - \frac{953.03}{38397.35} \approx \boxed{0.975}
\end{gather*}
</div>
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<div class="autopermalink" data-description="Solution 2.1"><a href="#quiz-05-4-1-2" title="Copy heading and permalink for Solution 2.1" aria-label="Copy heading and permalink for Solution 2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#quiz-05-4-1" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 5"><a href="#quiz-05" title="Copy heading and permalink for Worksheet: Quiz 5" aria-label="Copy heading and permalink for Worksheet: Quiz 5">🔗</a></div></section>
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