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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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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="Exam-1-Review"><h2 class="heading hide-type">
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</h2>
<section class="introduction" id="Exam-1-Review-2"><div class="para" id="Exam-1-Review-2-1">Use the following problems to prepare for the exam. There will be in-class review on Thursday, February 12. Your recitation this week will also be exam review.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="Exam-1-Review-3"><h3 class="heading hide-type">
<span class="type">Subsection</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">Allowed Materials</span>
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<div class="para" id="Exam-1-Review-3-2">You will be allowed to use a scientific calculator (<em class="emphasis">not</em> a graphing calculator, <em class="emphasis">not</em> a calculator app on your phone). You may not share a calculator with another student; you must use your own calculator.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="Exam-1-Review-3-3">You may bring a standard 3 in x 5 in index card with prepared notes. You may use both sides of the notecard. You must put your full name in the top right corner of the card, and turn it in along with your exam.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-3-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<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>
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<article class="exercise exercise-like" id="Exam-1-Review-4-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Exam-1-Review-4-1-1"><div class="para" id="Exam-1-Review-4-1-1-1">Consider the sets <span class="process-math">\(A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\text{,}\)</span> <span class="process-math">\(B = \{2, 4, 9, 10, 12, 14, 19\}\text{,}\)</span> and <span class="process-math">\(C = \{9, 10, 11, 14, 16, 17, 20\}\text{,}\)</span> which are all subsets of <span class="process-math">\(\Omega = \{1, 2, 3, \dotsc, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-1-1" 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="Exam-1-Review-4-1-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Exam-1-Review-4-1-2-1-1">Find <span class="process-math">\(A - (B \cap C)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-1-2-2" 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="Exam-1-Review-4-1-2-2-1">
<span class="process-math">\(\{1, 2, 3, 4, 5, 6, 7, 8\}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.a.1"><a href="#Exam-1-Review-4-1-2-2" title="Copy heading and permalink for Answer 1.a.1" aria-label="Copy heading and permalink for Answer 1.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.a"><a href="#Exam-1-Review-4-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-1-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Exam-1-Review-4-1-3-1-1">Find <span class="process-math">\(|A|\text{,}\)</span> <span class="process-math">\(|B|\text{,}\)</span> <span class="process-math">\(|C|\text{,}\)</span> <span class="process-math">\(|A\cup B|\text{,}\)</span> <span class="process-math">\(|A \cap B|\text{,}\)</span> <span class="process-math">\(|B\cap C|\text{,}\)</span> <span class="process-math">\(|A\cap C|\text{,}\)</span> and <span class="process-math">\(|A\cup B\cup C|\text{.}\)</span> Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-1-3-2" 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="Exam-1-Review-4-1-3-2-1">
<span class="process-math">\(|A| = 10\text{,}\)</span> <span class="process-math">\(|B| = 7\text{,}\)</span> <span class="process-math">\(|C| = 7\text{,}\)</span> <span class="process-math">\(|A \cup B| = 13\text{,}\)</span> <span class="process-math">\(|A \cap B| = 4\text{,}\)</span> <span class="process-math">\(|B\cap C| = 3\text{,}\)</span> <span class="process-math">\(|A\cap C| = 2\text{,}\)</span> <span class="process-math">\(|A\cup B\cup C| = 17\text{.}\)</span> In particular, note that <span class="process-math">\(|A\cup B| = 13 \neq 10 + 7 = |A| + |B|\text{,}\)</span> so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.b.1"><a href="#Exam-1-Review-4-1-3-2" title="Copy heading and permalink for Answer 1.b.1" aria-label="Copy heading and permalink for Answer 1.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.b"><a href="#Exam-1-Review-4-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-1-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Exam-1-Review-4-1-4-1-1">Find <span class="process-math">\(A^c\)</span> and <span class="process-math">\((A\cup B)^c\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-1-4-2" 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="Exam-1-Review-4-1-4-2-1">
<span class="process-math">\(A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\text{,}\)</span> <span class="process-math">\((A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.c.1"><a href="#Exam-1-Review-4-1-4-2" title="Copy heading and permalink for Answer 1.c.1" aria-label="Copy heading and permalink for Answer 1.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.c"><a href="#Exam-1-Review-4-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1"><a href="#Exam-1-Review-4-1" title="Copy heading and permalink for Exercise 1" aria-label="Copy heading and permalink for Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-2-1-1">Suppose we have a 6-sided die that’s weighted to roll a 6 half of the time. We roll the die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <span class="process-math">\((4, 2)\)</span>---rolling a 4 and then a 2.]<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-2-2" 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 logical" id="Exam-1-Review-4-2-2-1">
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-sample-space.html" id="Exam-1-Review-4-2-2-1-1">
\begin{align*}
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
\end{align*}
</div>
<div class="para">Note that <span class="process-math">\(\Omega\)</span> simply lists outcomes with no reference to the probabilities. So the answer here is the same as in <a href="notes-01-13.html#example-sample-space" class="xref" data-knowl="./knowl/xref/example-sample-space.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 4">Example 4</a>.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.1"><a href="#Exam-1-Review-4-2-2" title="Copy heading and permalink for Answer 2.1" aria-label="Copy heading and permalink for Answer 2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2"><a href="#Exam-1-Review-4-2" title="Copy heading and permalink for Exercise 2" aria-label="Copy heading and permalink for Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-3-1-1">Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-3-2" 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="Exam-1-Review-4-3-2-1">For two flips: <span class="process-math">\(\Omega = \{ HH, HT, TH, TT \}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Exam-1-Review-4-3-2-2">For three flips: <span class="process-math">\(\Omega = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-3-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="Exam-1-Review-4-3-2-3">
<div class="para">For four flips:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-3-2-3-1">
\begin{align*}
\Omega = \{ \amp HHHH, HHHT, HHTH, HTHH, \\
\amp THHH, HHTT, HTHT, HTTH, \\
\amp THHT, THTH, TTHH, HTTT, \\
\amp THTT, TTHT, TTTH, TTTT \}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-3-2-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Exam-1-Review-4-3-2-4">Each additional flip doubles the number of outcomes. So, with ten flips, we’ll have <span class="process-math">\(|\Omega| = 2^{10} = 1024\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-3-2-4" 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"><a href="#Exam-1-Review-4-3-2" title="Copy heading and permalink for Answer 3.1" aria-label="Copy heading and permalink for Answer 3.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 3"><a href="#Exam-1-Review-4-3" title="Copy heading and permalink for Exercise 3" aria-label="Copy heading and permalink for Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-4-1-1">If we roll a 6-sided die ten times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-4-2" 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="Exam-1-Review-4-4-2-1">Each additional roll will multiply the number of outcomes by 6. So, with 10 rolls, we’ll have <span class="process-math">\(|\Omega| = 6^{10}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 4.1"><a href="#Exam-1-Review-4-4-2" title="Copy heading and permalink for Answer 4.1" aria-label="Copy heading and permalink for Answer 4.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 4"><a href="#Exam-1-Review-4-4" title="Copy heading and permalink for Exercise 4" aria-label="Copy heading and permalink for Exercise 4">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-5-1-1">Consider the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\)</span> with probability distribution below. Calculate the probabilities of <span class="process-math">\(A = \{1, 3, 7, 8\}\text{,}\)</span> <span class="process-math">\(B = \{2, 3, 6, 7\}\text{,}\)</span> <span class="process-math">\(A\cup B\text{,}\)</span> and <span class="process-math">\(A \cap B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-1-Review-4-5-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">93<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 93: "><a href="#Exam-1-Review-4-5-1-2" title="Copy heading and permalink for Table 93: " aria-label="Copy heading and permalink for Table 93: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</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="solutions"><details id="Exam-1-Review-4-5-2" 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="Exam-1-Review-4-5-2-1">
<span class="process-math">\(\Pr(A) = 0.45, \Pr(B) = 0.4, \Pr(A \cup B) = 0.6, \Pr(A \cap B) = 0.25.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 5.1"><a href="#Exam-1-Review-4-5-2" title="Copy heading and permalink for Answer 5.1" aria-label="Copy heading and permalink for Answer 5.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 5"><a href="#Exam-1-Review-4-5" title="Copy heading and permalink for Exercise 5" aria-label="Copy heading and permalink for Exercise 5">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-6-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-6-2" 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">
<figure class="table table-like" id="Exam-1-Review-4-6-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">94<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for a Linearly Scaled Die<div class="autopermalink" data-description="Table 94: Probability Distribution for a Linearly Scaled Die"><a href="#Exam-1-Review-4-6-2-1" title="Copy heading and permalink for Table 94: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 94: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/21\)</span></td>
</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">\(2/21\)</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">\(3/21\)</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">\(4/21\)</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">\(5/21\)</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">\(6/21\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 6.1"><a href="#Exam-1-Review-4-6-2" title="Copy heading and permalink for Answer 6.1" aria-label="Copy heading and permalink for Answer 6.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 6"><a href="#Exam-1-Review-4-6" title="Copy heading and permalink for Exercise 6" aria-label="Copy heading and permalink for Exercise 6">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-7-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-7-2" 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">
<figure class="table table-like" id="Exam-1-Review-4-7-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">95<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for an Even-biased Die<div class="autopermalink" data-description="Table 95: Probability Distribution for an Even-biased Die"><a href="#Exam-1-Review-4-7-2-1" title="Copy heading and permalink for Table 95: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 95: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/9\)</span></td>
</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">\(2/9\)</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">\(1/9\)</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">\(2/9\)</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">\(1/9\)</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">\(2/9\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 7.1"><a href="#Exam-1-Review-4-7-2" title="Copy heading and permalink for Answer 7.1" aria-label="Copy heading and permalink for Answer 7.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 7"><a href="#Exam-1-Review-4-7" title="Copy heading and permalink for Exercise 7" aria-label="Copy heading and permalink for Exercise 7">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-8-1-1">A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> find the probability of the toxin molecule leaving the cell during the <span class="process-math">\(n\)</span>th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-8-2" 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="Exam-1-Review-4-8-2-1">For short, write <span class="process-math">\(\Pr(n)\)</span> to mean the probability of the toxin molecule leaving during the <span class="process-math">\(n\)</span>th minute. Then <span class="process-math">\(\Pr(n) = (0.7)^{n - 1} (0.3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Exam-1-Review-4-8-2-2">The probability of leaving during the first 3 minutes is <span class="process-math">\(\Pr(1) + \Pr(2) + \Pr(3) = 0.657.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-8-2-2" 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="Answer 8.1"><a href="#Exam-1-Review-4-8-2" title="Copy heading and permalink for Answer 8.1" aria-label="Copy heading and permalink for Answer 8.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 8"><a href="#Exam-1-Review-4-8" title="Copy heading and permalink for Exercise 8" aria-label="Copy heading and permalink for Exercise 8">🔗</a></div></article><div class="exercisegroup" id="Exam-1-Review-4-9">
<h4 class="heading"><span class="title">Exercise Group.</span></h4>
<div class="introduction" id="Exam-1-Review-4-9-1"><div class="para" id="Exam-1-Review-4-9-1-1">In each of the following scenarios with given events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> alculate <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> <span class="process-math">\(\Pr(A\cap B)\text{,}\)</span> <span class="process-math">\(\Pr(A \mid B)\text{,}\)</span> and <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<div class="exercisegroup-exercises">
<article class="exercise exercise-like" id="Exam-1-Review-4-9-2"><h5 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
<div class="para" id="Exam-1-Review-4-9-2-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-9-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-9-2-2" 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 logical" id="Exam-1-Review-4-9-2-2-1">
<div class="displaymath process-math" id="Exam-1-Review-4-9-2-2-1-1">
\begin{align*}
A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 4), (3, 5), (3, 6), \\
\amp (4, 5), (4, 6), \\
\amp (5, 6)\} \\
A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-9-2-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{6/36}{18/36} = \frac{6}{18} = \frac{1}{3}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-9-2-2-1" 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="Answer 9.1"><a href="#Exam-1-Review-4-9-2-2" title="Copy heading and permalink for Answer 9.1" aria-label="Copy heading and permalink for Answer 9.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 9"><a href="#Exam-1-Review-4-9-2" title="Copy heading and permalink for Exercise 9" aria-label="Copy heading and permalink for Exercise 9">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-9-3"><h5 class="heading"><span class="codenumber">10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
<div class="para" id="Exam-1-Review-4-9-3-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-9-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-9-3-2" 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 logical" id="Exam-1-Review-4-9-3-2-1">
<div class="displaymath process-math" id="Exam-1-Review-4-9-3-2-1-1">
\begin{align*}
A \amp = \{ HHH, HHT, TTH, TTT \} \\
B \amp = \{ HHH, HHT, HTH, THH \} \\
A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-9-3-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2}
\end{align*}
</div>
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<div class="autopermalink" data-description="Answer 10.1"><a href="#Exam-1-Review-4-9-3-2" title="Copy heading and permalink for Answer 10.1" aria-label="Copy heading and permalink for Answer 10.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 10"><a href="#Exam-1-Review-4-9-3" title="Copy heading and permalink for Exercise 10" aria-label="Copy heading and permalink for Exercise 10">🔗</a></div></article>
</div>
<div class="autopermalink" data-description="Exercise Group 9–10"><a href="#Exam-1-Review-4-9" title="Copy heading and permalink for Exercise Group 9–10" aria-label="Copy heading and permalink for Exercise Group 9–10">🔗</a></div>
</div>
<article class="exercise exercise-like" id="Exam-1-Review-4-10"><h4 class="heading"><span class="codenumber">11<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Exam-1-Review-4-10-1"><div class="para" id="Exam-1-Review-4-10-1-1">A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-10-1-1" 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="Exam-1-Review-4-10-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Exam-1-Review-4-10-2-1-1">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-10-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-10-2-2" 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 logical" id="Exam-1-Review-4-10-2-2-1">
<div class="para">Let <span class="process-math">\(P\)</span> be the event of testing positive and <span class="process-math">\(D\)</span> the event of having the disease. Then the prevalence <span class="process-math">\(\Pr(D)\)</span> is given as 3.2%, or 0.032. The sensitivity is <span class="process-math">\(\Pr(P\mid D) = 0.65\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c\mid D^c) = 0.999\text{.}\)</span> So, according to Bayes’ Theorem:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-10-2-2-1-6">
\begin{align*}
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\mid D^c))\Pr(D^c)} \\
\amp = \frac{(0.65)(0.032)}{(0.65)(0.032) + (1 - 0.999)(1 - 0.032)} \\
\amp \approx 0.96
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-10-2-2-1" 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="Answer 11.a.1"><a href="#Exam-1-Review-4-10-2-2" title="Copy heading and permalink for Answer 11.a.1" aria-label="Copy heading and permalink for Answer 11.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 11.a"><a href="#Exam-1-Review-4-10-2" title="Copy heading and permalink for Task 11.a" aria-label="Copy heading and permalink for Task 11.a">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-10-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Exam-1-Review-4-10-3-1-1">For a patient who receives a negative test, what is the probability they do not have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-10-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-10-3-2" 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 logical" id="Exam-1-Review-4-10-3-2-1">
<div class="displaymath process-math" id="Exam-1-Review-4-10-3-2-1-1">
\begin{align*}
\Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + (1 - \Pr(P\mid D))\Pr(D)} \\
\amp = \frac{(0.999)(1 - 0.032)}{(0.999)(1 - 0.032) + (1 - 0.65)(0.032)} \\
\amp \approx 0.99
\end{align*}
</div>
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<div class="autopermalink" data-description="Answer 11.b.1"><a href="#Exam-1-Review-4-10-3-2" title="Copy heading and permalink for Answer 11.b.1" aria-label="Copy heading and permalink for Answer 11.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 11.b"><a href="#Exam-1-Review-4-10-3" title="Copy heading and permalink for Task 11.b" aria-label="Copy heading and permalink for Task 11.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 11"><a href="#Exam-1-Review-4-10" title="Copy heading and permalink for Exercise 11" aria-label="Copy heading and permalink for Exercise 11">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-11"><h4 class="heading"><span class="codenumber">12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-11-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-11-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-11-2" 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 logical" id="Exam-1-Review-4-11-2-1">
<div class="displaymath process-math" id="Exam-1-Review-4-11-2-1-1">
\begin{align*}
A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 4), (3, 5), (3, 6), \\
\amp (4, 5), (4, 6), \\
\amp (5, 6)\} \\
A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-11-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \neq \Pr(A),
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.</div>
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<div class="autopermalink" data-description="Answer 12.1"><a href="#Exam-1-Review-4-11-2" title="Copy heading and permalink for Answer 12.1" aria-label="Copy heading and permalink for Answer 12.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 12"><a href="#Exam-1-Review-4-11" title="Copy heading and permalink for Exercise 12" aria-label="Copy heading and permalink for Exercise 12">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-12"><h4 class="heading"><span class="codenumber">13<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-12-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-12-2" 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 logical" id="Exam-1-Review-4-12-2-1">
<div class="displaymath process-math" id="Exam-1-Review-4-12-2-1-1">
\begin{align*}
A \amp = \{ HHH, HHT, TTH, TTT \} \\
B \amp = \{ HHH, HHT, HTH, THH \} \\
A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-12-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} = \Pr(A),
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<div class="autopermalink" data-description="Answer 13.1"><a href="#Exam-1-Review-4-12-2" title="Copy heading and permalink for Answer 13.1" aria-label="Copy heading and permalink for Answer 13.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 13"><a href="#Exam-1-Review-4-12" title="Copy heading and permalink for Exercise 13" aria-label="Copy heading and permalink for Exercise 13">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-13"><h4 class="heading"><span class="codenumber">14<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-13-1-1">Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{3, 4, 5\}\)</span> be events in the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> Create a probability distribution for <span class="process-math">\(\Omega\)</span> so that <span class="process-math">\(A, B\)</span> are independent.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-13-2" 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="Exam-1-Review-4-13-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">96<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 96: Example Distribution"><a href="#Exam-1-Review-4-13-2-1" title="Copy heading and permalink for Table 96: Example Distribution" aria-label="Copy heading and permalink for Table 96: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</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">0.2</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">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure><div class="para logical" id="Exam-1-Review-4-13-2-2">
<div class="para">Now <span class="process-math">\(\Pr(A) = 0.5\text{,}\)</span> <span class="process-math">\(\Pr(B) = 0.4\text{,}\)</span> and</div>
<div class="displaymath process-math" id="Exam-1-Review-4-13-2-2-3">
\begin{gather*}
\Pr(A\cap B) = 0.2 = (0.5)(0.4) = \Pr(A)\Pr(B),
\end{gather*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-13-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 14.1"><a href="#Exam-1-Review-4-13-2" title="Copy heading and permalink for Answer 14.1" aria-label="Copy heading and permalink for Answer 14.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 14"><a href="#Exam-1-Review-4-13" title="Copy heading and permalink for Exercise 14" aria-label="Copy heading and permalink for Exercise 14">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-14"><h4 class="heading"><span class="codenumber">15<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-14-1-1">An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability <span class="process-math">\(p = 0.3\text{,}\)</span> find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-14-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-14-2" 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="Exam-1-Review-4-14-2-1">
<div class="para">Let <span class="process-math">\(S\)</span> be the number of heads. Then <span class="process-math">\(S \sim \Bin(20, 0.3)\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-14-2-1-3">
\begin{align*}
\Pr(S = 5) \amp = {20 \choose 5} (0.3)^5 (0.7)^{20 - 5} \\
\amp = \frac{20!}{(5!)(15!)} (0.3)^5 (0.7)^{15} \\
\amp = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} (0.3)^5 (0.7)^{15} \\
\amp = (19 \times 3 \times 17 \times 16) (0.3)^5 (0.7)^{15} \\
\amp \approx 0.179
\end{align*}
</div>
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<div class="autopermalink" data-description="Solution 15.1"><a href="#Exam-1-Review-4-14-2" title="Copy heading and permalink for Solution 15.1" aria-label="Copy heading and permalink for Solution 15.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 15"><a href="#Exam-1-Review-4-14" title="Copy heading and permalink for Exercise 15" aria-label="Copy heading and permalink for Exercise 15">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-15"><h4 class="heading"><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Exam-1-Review-4-15-1"><div class="para" id="Exam-1-Review-4-15-1-1">An experiment consists of flipping a coin repeatedly until we first see heads.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-15-1-1" 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="Exam-1-Review-4-15-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Exam-1-Review-4-15-2-1-1">If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-15-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-15-2-2" 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="Exam-1-Review-4-15-2-2-1">
<div class="para">Let <span class="process-math">\(T\)</span> be the number of flips until we see heads. Then <span class="process-math">\(T\)</span> is geometric with parameter <span class="process-math">\(p = 0.4\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-15-2-2-1-4">
\begin{align*}
\Pr(T = k) \amp = (1-0.4)^{k-1}(0.4) = 0.6^{k-1} \cdot 0.4
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-15-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 16.a.1"><a href="#Exam-1-Review-4-15-2-2" title="Copy heading and permalink for Solution 16.a.1" aria-label="Copy heading and permalink for Solution 16.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 16.a"><a href="#Exam-1-Review-4-15-2" title="Copy heading and permalink for Task 16.a" aria-label="Copy heading and permalink for Task 16.a">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-15-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Exam-1-Review-4-15-3-1-1">Which flip has the highest chance of being the first flip to come up heads?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-15-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 16.b"><a href="#Exam-1-Review-4-15-3" title="Copy heading and permalink for Task 16.b" aria-label="Copy heading and permalink for Task 16.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 16"><a href="#Exam-1-Review-4-15" title="Copy heading and permalink for Exercise 16" aria-label="Copy heading and permalink for Exercise 16">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-16"><h4 class="heading"><span class="codenumber">17<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-16-1-1">A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-16-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 17"><a href="#Exam-1-Review-4-16" title="Copy heading and permalink for Exercise 17" aria-label="Copy heading and permalink for Exercise 17">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-17"><h4 class="heading"><span class="codenumber">18<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Exam-1-Review-4-17-1"><div class="para" id="Exam-1-Review-4-17-1-1">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><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-17-1-1" 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="Exam-1-Review-4-17-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Exam-1-Review-4-17-2-1-1">What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-17-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-17-2-2" 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="Exam-1-Review-4-17-2-2-1">
<span class="process-math">\(k = \frac{6}{17}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-17-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 18.a.1"><a href="#Exam-1-Review-4-17-2-2" title="Copy heading and permalink for Answer 18.a.1" aria-label="Copy heading and permalink for Answer 18.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 18.a"><a href="#Exam-1-Review-4-17-2" title="Copy heading and permalink for Task 18.a" aria-label="Copy heading and permalink for Task 18.a">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-17-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Exam-1-Review-4-17-3-1-1">Find <span class="process-math">\(\Pr(2 \leq X \leq 3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-17-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-17-3-2" 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="Exam-1-Review-4-17-3-2-1">
<span class="process-math">\(\Pr(2 \leq X \leq 3) \approx 0.325\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-17-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 18.b.1"><a href="#Exam-1-Review-4-17-3-2" title="Copy heading and permalink for Answer 18.b.1" aria-label="Copy heading and permalink for Answer 18.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 18.b"><a href="#Exam-1-Review-4-17-3" title="Copy heading and permalink for Task 18.b" aria-label="Copy heading and permalink for Task 18.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 18"><a href="#Exam-1-Review-4-17" title="Copy heading and permalink for Exercise 18" aria-label="Copy heading and permalink for Exercise 18">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-18"><h4 class="heading"><span class="codenumber">19<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-18-1-1">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 the c.d.f. <span class="process-math">\(F(x)\text{.}\)</span> Use your c.d.f. to find <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-18-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-18-2" 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="Exam-1-Review-4-18-2-1">
<span class="process-math">\(\frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}\text{.}\)</span> <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-18-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 19.1"><a href="#Exam-1-Review-4-18-2" title="Copy heading and permalink for Answer 19.1" aria-label="Copy heading and permalink for Answer 19.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 19"><a href="#Exam-1-Review-4-18" title="Copy heading and permalink for Exercise 19" aria-label="Copy heading and permalink for Exercise 19">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-19"><h4 class="heading"><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-19-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([2, 3]\)</span> has c.d.f. <span class="process-math">\(F(x) = \frac{x^3}{3} - x^2 + 4\text{.}\)</span> Find the p.d.f. <span class="process-math">\(f(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-19-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-19-2" 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="Exam-1-Review-4-19-2-1">
<span class="process-math">\(f(x) = x^2 - 2x.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-19-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 20.1"><a href="#Exam-1-Review-4-19-2" title="Copy heading and permalink for Answer 20.1" aria-label="Copy heading and permalink for Answer 20.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 20"><a href="#Exam-1-Review-4-19" title="Copy heading and permalink for Exercise 20" aria-label="Copy heading and permalink for Exercise 20">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-20"><h4 class="heading"><span class="codenumber">21<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-20-1-1">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="#Exam-1-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-1-Review-4-20-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">97<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 97: Joint distribution for X, Y"><a href="#Exam-1-Review-4-20-1-2" title="Copy heading and permalink for Table 97: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 97: 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="Exam-1-Review-4-20-2" 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="Exam-1-Review-4-20-2-1">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="#Exam-1-Review-4-20-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 21.1"><a href="#Exam-1-Review-4-20-2" title="Copy heading and permalink for Answer 21.1" aria-label="Copy heading and permalink for Answer 21.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 21"><a href="#Exam-1-Review-4-20" title="Copy heading and permalink for Exercise 21" aria-label="Copy heading and permalink for Exercise 21">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-21"><h4 class="heading"><span class="codenumber">22<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para logical" id="Exam-1-Review-4-21-1-1">
<div class="para">Suppose <span class="process-math">\(X, Y\)</span> have the distributions:</div>
<div class="displaymath process-math" id="Exam-1-Review-4-21-1-1-2">
\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="#Exam-1-Review-4-21-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-21-2" 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="Exam-1-Review-4-21-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">98<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 98: Joint Distribution"><a href="#Exam-1-Review-4-21-2-1" title="Copy heading and permalink for Table 98: Joint Distribution" aria-label="Copy heading and permalink for Table 98: 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 22.1"><a href="#Exam-1-Review-4-21-2" title="Copy heading and permalink for Answer 22.1" aria-label="Copy heading and permalink for Answer 22.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 22"><a href="#Exam-1-Review-4-21" title="Copy heading and permalink for Exercise 22" aria-label="Copy heading and permalink for Exercise 22">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-22"><h4 class="heading"><span class="codenumber">23<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-22-1-1">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="#Exam-1-Review-4-22-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-1-Review-4-22-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">99<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 99: "><a href="#Exam-1-Review-4-22-1-2" title="Copy heading and permalink for Table 99: " aria-label="Copy heading and permalink for Table 99: ">🔗</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="solutions"><details id="Exam-1-Review-4-22-2" 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="Exam-1-Review-4-22-2-1">4.8.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-22-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 23.1"><a href="#Exam-1-Review-4-22-2" title="Copy heading and permalink for Answer 23.1" aria-label="Copy heading and permalink for Answer 23.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 23"><a href="#Exam-1-Review-4-22" title="Copy heading and permalink for Exercise 23" aria-label="Copy heading and permalink for Exercise 23">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-23"><h4 class="heading"><span class="codenumber">24<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Exam-1-Review-4-23-1"><div class="para" id="Exam-1-Review-4-23-1-1">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="#Exam-1-Review-4-23-1-1" 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="Exam-1-Review-4-23-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Exam-1-Review-4-23-2-1-1">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="#Exam-1-Review-4-23-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-23-2-2" 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="Exam-1-Review-4-23-2-2-1">40.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-23-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.a.1"><a href="#Exam-1-Review-4-23-2-2" title="Copy heading and permalink for Answer 24.a.1" aria-label="Copy heading and permalink for Answer 24.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.a"><a href="#Exam-1-Review-4-23-2" title="Copy heading and permalink for Task 24.a" aria-label="Copy heading and permalink for Task 24.a">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-23-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Exam-1-Review-4-23-3-1-1">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="#Exam-1-Review-4-23-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-23-3-2" 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="Exam-1-Review-4-23-3-2-1">48.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-23-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.b.1"><a href="#Exam-1-Review-4-23-3-2" title="Copy heading and permalink for Answer 24.b.1" aria-label="Copy heading and permalink for Answer 24.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.b"><a href="#Exam-1-Review-4-23-3" title="Copy heading and permalink for Task 24.b" aria-label="Copy heading and permalink for Task 24.b">🔗</a></div></article><article class="task exercise-like" id="Exam-1-Review-4-23-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Exam-1-Review-4-23-4-1-1">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="#Exam-1-Review-4-23-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-23-4-2" 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="Exam-1-Review-4-23-4-2-1">100.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-23-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.c.1"><a href="#Exam-1-Review-4-23-4-2" title="Copy heading and permalink for Answer 24.c.1" aria-label="Copy heading and permalink for Answer 24.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.c"><a href="#Exam-1-Review-4-23-4" title="Copy heading and permalink for Task 24.c" aria-label="Copy heading and permalink for Task 24.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 24"><a href="#Exam-1-Review-4-23" title="Copy heading and permalink for Exercise 24" aria-label="Copy heading and permalink for Exercise 24">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-24"><h4 class="heading"><span class="codenumber">25<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-24-1-1">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="#Exam-1-Review-4-24-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-24-2" 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="Exam-1-Review-4-24-2-1">4.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-24-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 25.1"><a href="#Exam-1-Review-4-24-2" title="Copy heading and permalink for Answer 25.1" aria-label="Copy heading and permalink for Answer 25.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 25"><a href="#Exam-1-Review-4-24" title="Copy heading and permalink for Exercise 25" aria-label="Copy heading and permalink for Exercise 25">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-25"><h4 class="heading"><span class="codenumber">26<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-25-1-1">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="#Exam-1-Review-4-25-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-25-2" 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="Exam-1-Review-4-25-2-1">2/3.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-25-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 26.1"><a href="#Exam-1-Review-4-25-2" title="Copy heading and permalink for Answer 26.1" aria-label="Copy heading and permalink for Answer 26.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 26"><a href="#Exam-1-Review-4-25" title="Copy heading and permalink for Exercise 26" aria-label="Copy heading and permalink for Exercise 26">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-26"><h4 class="heading"><span class="codenumber">27<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-26-1-1">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="#Exam-1-Review-4-26-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-26-2" 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="Exam-1-Review-4-26-2-1">
<span class="process-math">\(\frac{4}{3}\ln(4) \approx 1.85.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-26-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 27.1"><a href="#Exam-1-Review-4-26-2" title="Copy heading and permalink for Answer 27.1" aria-label="Copy heading and permalink for Answer 27.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 27"><a href="#Exam-1-Review-4-26" title="Copy heading and permalink for Exercise 27" aria-label="Copy heading and permalink for Exercise 27">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-27"><h4 class="heading"><span class="codenumber">28<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-27-1-1">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="#Exam-1-Review-4-27-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Exam-1-Review-4-27-2" 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="Exam-1-Review-4-27-2-1">
<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="#Exam-1-Review-4-27-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 28.1"><a href="#Exam-1-Review-4-27-2" title="Copy heading and permalink for Answer 28.1" aria-label="Copy heading and permalink for Answer 28.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 28"><a href="#Exam-1-Review-4-27" title="Copy heading and permalink for Exercise 28" aria-label="Copy heading and permalink for Exercise 28">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-28"><h4 class="heading"><span class="codenumber">29<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-28-1-1">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="#Exam-1-Review-4-28-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-1-Review-4-28-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">100<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 100: "><a href="#Exam-1-Review-4-28-1-2" title="Copy heading and permalink for Table 100: " aria-label="Copy heading and permalink for Table 100: ">🔗</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 29"><a href="#Exam-1-Review-4-28" title="Copy heading and permalink for Exercise 29" aria-label="Copy heading and permalink for Exercise 29">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-29"><h4 class="heading"><span class="codenumber">30<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-29-1-1">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="#Exam-1-Review-4-29-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 30"><a href="#Exam-1-Review-4-29" title="Copy heading and permalink for Exercise 30" aria-label="Copy heading and permalink for Exercise 30">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-30"><h4 class="heading"><span class="codenumber">31<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-30-1-1">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="#Exam-1-Review-4-30-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 31"><a href="#Exam-1-Review-4-30" title="Copy heading and permalink for Exercise 31" aria-label="Copy heading and permalink for Exercise 31">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-31"><h4 class="heading"><span class="codenumber">32<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-1-Review-4-32-1-1">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="#Exam-1-Review-4-32-1-1" 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 33"><a href="#Exam-1-Review-4-32" title="Copy heading and permalink for Exercise 33" aria-label="Copy heading and permalink for Exercise 33">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-33"><h4 class="heading"><span class="codenumber">34<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-1-Review-4-33-1-1">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="#Exam-1-Review-4-33-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 34"><a href="#Exam-1-Review-4-33" title="Copy heading and permalink for Exercise 34" aria-label="Copy heading and permalink for Exercise 34">🔗</a></div></article><div class="autopermalink" data-description="Exercises"><a href="#Exam-1-Review-4" title="Copy heading and permalink for Exercises" aria-label="Copy heading and permalink for Exercises">🔗</a></div></section><div class="autopermalink" data-description="Section: Exam 1 Review"><a href="#Exam-1-Review" title="Copy heading and permalink for Section: Exam 1 Review" aria-label="Copy heading and permalink for Section: Exam 1 Review">🔗</a></div></section></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
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<li><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></li>
<li><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></li>
<li><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></li>
<li><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></li>
</ul></nav><div class="autopermalink" data-description="Chapter: Class Notes"><a href="#ch-notes" title="Copy heading and permalink for Chapter: Class Notes" aria-label="Copy heading and permalink for Chapter: Class Notes">🔗</a></div></section></div>
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+15 -1
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@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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<li class="toc-item toc-chapter">
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<span class="title">Introduction to Probability and Statistics for the Life Sciences</span><span class="subtitle">Spring 2026 MATH 1044</span>
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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<li class="toc-item toc-chapter">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="review.html" class="internal"><span class="title">Exam Review</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section">
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eBookConfig.runestone_version = '7.11.13';
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@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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+17 -3
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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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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-03"><h2 class="heading hide-type">
<span class="type">Section</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">Tuesday, Feb 3</span>
</h2>
<section class="introduction" id="notes-02-03-2"><div class="para" id="notes-02-03-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="notes-02-03-3"><h3 class="heading hide-type">
<span class="type">Subsection</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">Variance</span>
</h3>
<div class="para logical" id="notes-02-03-3-2">
<div class="para">Question: How spread out are <span class="process-math">\(X\)</span> values? One answer we might try is to measure the average distance from the average value:</div>
<div class="displaymath process-math" id="notes-02-03-3-2-2">
\begin{gather*}
\E\left[|X - \E(X)|\right]
\end{gather*}
</div>
<div class="para">To simplify notation, we’ll write <span class="process-math">\(\mu = \E(X)\text{.}\)</span> Also, it’s often usefull to square a term rather than take absolute value when we want to ensure a positive output, so we’ll define...</div>
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</div>
<article class="definition definition-like" id="def-variance"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="def-variance-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable with <span class="process-math">\(\E(X) = \mu\text{.}\)</span> The <dfn class="terminology">variance</dfn> of <span class="process-math">\(X\)</span> is:</div>
<div class="displaymath process-math" id="def-variance-1-1-5">
\begin{gather*}
\sigma^2 = \Var(x) = \E\left[ (X - \mu)^2\right]
\end{gather*}
</div>
<div class="para">
<span class="process-math">\(\sigma = \sqrt{\Var(X)}\)</span> is called the <dfn class="terminology">standard deviation</dfn>.</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 90"><a href="#def-variance" title="Copy heading and permalink for Definition 90" aria-label="Copy heading and permalink for Definition 90">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Variance"><a href="#notes-02-03-3" title="Copy heading and permalink for Subsection: Variance" aria-label="Copy heading and permalink for Subsection: Variance">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Feb 3"><a href="#notes-02-03" title="Copy heading and permalink for Section: Tuesday, Feb 3" aria-label="Copy heading and permalink for Section: Tuesday, Feb 3">🔗</a></div></section></div>
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+15 -1
View File
@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter contains-active">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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+15 -1
View File
@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
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<li class="toc-item toc-chapter contains-active">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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+30 -4
View File
@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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</ul>
</li>
<li class="toc-item toc-chapter contains-active">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content">
<div class="print-preview-header">
@@ -281,7 +295,7 @@ eBookConfig.enable_chatcodes = false;
<div class="introduction" id="quiz-02-3-2-1">
<div class="para" id="quiz-02-3-2-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-1-1" 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-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90<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 90: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 90: Joint Distribution" aria-label="Copy heading and permalink for Table 90: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91<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 91: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 91: Joint Distribution" aria-label="Copy heading and permalink for Table 91: 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 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
@@ -322,8 +336,20 @@ eBookConfig.enable_chatcodes = false;
<div class="para" id="quiz-02-3-2-3-1-1">Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="quiz-02-3-2-3-2" 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" id="quiz-02-3-2-3-2-1">
<span class="process-math">\(\Pr(X = 2, Y = 0) = 0.2 \text{,}\)</span> and <span class="process-math">\(\Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1\text{.}\)</span> Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para logical" id="quiz-02-3-2-3-2-1">
<div class="para">Checking each cell in the table:</div>
<div class="displaymath process-math" id="quiz-02-3-2-3-2-1-1">
\begin{align*}
\Pr(X = 0, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 0)\Pr(Y = 0) \\
\Pr(X = 1, Y = 0) \amp = 0.05 = (0.2)(0.25) = \Pr(X = 1)\Pr(Y = 0) \\
\Pr(X = 2, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 2)\Pr(Y = 0) \\
\Pr(X = 0, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 0)\Pr(Y = 1) \\
\Pr(X = 1, Y = 1) \amp = 0.15 = (0.2)(0.75) = \Pr(X = 1)\Pr(Y = 1) \\
\Pr(X = 2, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 2)\Pr(Y = 1)
\end{align*}
</div>
<div class="para">So <span class="process-math">\(X, Y\)</span> are independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-02-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
</div></details></div>
+30 -4
View File
@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter contains-active">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<div class="toc-title-box"><a href="review.html" class="internal"><span class="title">Exam Review</span></a></div>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="worksheet" id="quiz-02" data-margins="0.75in 0.75in 0.75in 0.75in"><h2 class="heading hide-type">
<span class="type">Worksheet</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">Quiz 2</span><div class="print-links"><a href="quiz-02-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div>
@@ -266,7 +280,7 @@ eBookConfig.enable_chatcodes = false;
<div class="introduction" id="quiz-02-3-2-1">
<div class="para" id="quiz-02-3-2-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-1-1" 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-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90<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 90: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 90: Joint Distribution" aria-label="Copy heading and permalink for Table 90: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91<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 91: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 91: Joint Distribution" aria-label="Copy heading and permalink for Table 91: 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 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
@@ -307,8 +321,20 @@ eBookConfig.enable_chatcodes = false;
<div class="para" id="quiz-02-3-2-3-1-1">Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="quiz-02-3-2-3-2" 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" id="quiz-02-3-2-3-2-1">
<span class="process-math">\(\Pr(X = 2, Y = 0) = 0.2 \text{,}\)</span> and <span class="process-math">\(\Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1\text{.}\)</span> Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para logical" id="quiz-02-3-2-3-2-1">
<div class="para">Checking each cell in the table:</div>
<div class="displaymath process-math" id="quiz-02-3-2-3-2-1-1">
\begin{align*}
\Pr(X = 0, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 0)\Pr(Y = 0) \\
\Pr(X = 1, Y = 0) \amp = 0.05 = (0.2)(0.25) = \Pr(X = 1)\Pr(Y = 0) \\
\Pr(X = 2, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 2)\Pr(Y = 0) \\
\Pr(X = 0, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 0)\Pr(Y = 1) \\
\Pr(X = 1, Y = 1) \amp = 0.15 = (0.2)(0.75) = \Pr(X = 1)\Pr(Y = 1) \\
\Pr(X = 2, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 2)\Pr(Y = 1)
\end{align*}
</div>
<div class="para">So <span class="process-math">\(X, Y\)</span> are independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-02-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
</div></details></div>
+17 -3
View File
@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe14d;</span><span class="name">Copy</span></button>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter contains-active">
@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="chapter" id="quizzes"><h2 class="heading hide-type">
<span class="type">Chapter</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">Quizzes</span>
@@ -237,7 +251,7 @@ eBookConfig.enable_chatcodes = false;
<li><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></li>
</ul></nav><div class="autopermalink" data-description="Chapter: Quizzes"><a href="#quizzes" title="Copy heading and permalink for Chapter: Quizzes" aria-label="Copy heading and permalink for Chapter: Quizzes">🔗</a></div></section></div>
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@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.12';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe14d;</span><span class="name">Copy</span></button>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
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@@ -295,7 +309,7 @@ eBookConfig.enable_chatcodes = false;
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<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">92</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
</h3>
<div class="para logical" id="thm-FTC-2-1">
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
@@ -306,7 +320,7 @@ eBookConfig.enable_chatcodes = false;
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<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-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="Theorem 91: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 91: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 91: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Theorem 92: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
@@ -324,7 +338,7 @@ eBookConfig.enable_chatcodes = false;
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+19 -5
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@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
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eBookConfig.build_info = "";
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@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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@@ -228,6 +232,16 @@ eBookConfig.enable_chatcodes = false;
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@@ -280,7 +294,7 @@ eBookConfig.enable_chatcodes = false;
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<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">92</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
</h3>
<div class="para logical" id="thm-FTC-2-1">
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
@@ -291,7 +305,7 @@ eBookConfig.enable_chatcodes = false;
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<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-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="Theorem 91: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 91: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 91: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Theorem 92: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
@@ -308,7 +322,7 @@ eBookConfig.enable_chatcodes = false;
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@@ -101,7 +101,7 @@ eBookConfig.allow_pairs = false;
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@@ -215,6 +215,10 @@ eBookConfig.enable_chatcodes = false;
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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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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