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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<ul class="structural toc-item-list">
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<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
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<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></div>
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<li class="toc-item toc-section">
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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>
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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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|
||||
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|
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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>
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<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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|
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|
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|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
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<section class="worksheet" id="exam-01" data-margins="0.75in 0.75in 0.75in 0.75in"><h1 class="heading hide-type">
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<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">Exam 1</span><div class="print-links"><a href="exam-01-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></div>
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</h1>
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<section class="introduction" id="exam-01-2"><div class="para" id="exam-01-2-1">Show all relevant work.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="onepage" id="exam-01-3"><article class="exercise exercise-like" id="exam-01-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-3-1-1"><div class="para" id="exam-01-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div>
|
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<article class="task exercise-like" id="exam-01-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-2-1-1">Let <span class="process-math">\(A, B\)</span> be any sets. Then <span class="process-math">\(|A \cap B| = |A| \cdot |B|\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-01-3-1-2-2" 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" id="exam-01-3-1-2-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.a.1"><a href="#exam-01-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
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</div></details></div>
|
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<div class="autopermalink" data-description="Task 1.a"><a href="#exam-01-3-1-2" 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-01-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-3-1-1">Let <span class="process-math">\(A, B\)</span> be any events. Then <span class="process-math">\(\Pr(A \mid B) = \Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-3-2" 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="exam-01-3-1-3-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.b.1"><a href="#exam-01-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.b"><a href="#exam-01-3-1-3" 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-01-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-4-1-1">Given a joint distribution for random variables <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{,}\)</span> then it must be the case that <span class="process-math">\(\Pr(X = 0, Y = 0) = \Pr(X = 0)\Pr(Y = 0)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-4-2" 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="exam-01-3-1-4-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.c.1"><a href="#exam-01-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.c"><a href="#exam-01-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-5"><h4 class="heading"><span class="codenumber">(d)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-5-1-1">Suppose we flip a coin repeatedly until we first see heads and let <span class="process-math">\(T\)</span> be the number of flips. Then <span class="process-math">\(\Pr(T = 2) \geq \Pr(T = 4)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-5-2" 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="exam-01-3-1-5-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.d.1"><a href="#exam-01-3-1-5-2" title="Copy heading and permalink for Solution 1.d.1" aria-label="Copy heading and permalink for Solution 1.d.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.d"><a href="#exam-01-3-1-5" title="Copy heading and permalink for Task 1.d" aria-label="Copy heading and permalink for Task 1.d">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-6"><h4 class="heading"><span class="codenumber">(e)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-6-1-1">If molecules are observed leaving a cell after 1.5 minutes, 1.8 minutes, 2.1 minutes, 2.2 minutes, and 2.4 minutes, then the maximum likelihood estimation for the rate at which molecules leave the cell is <span class="process-math">\(1/2\)</span> per minute.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-6-2" 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="exam-01-3-1-6-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-6-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.e.1"><a href="#exam-01-3-1-6-2" title="Copy heading and permalink for Solution 1.e.1" aria-label="Copy heading and permalink for Solution 1.e.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.e"><a href="#exam-01-3-1-6" title="Copy heading and permalink for Task 1.e" aria-label="Copy heading and permalink for Task 1.e">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-7"><h4 class="heading"><span class="codenumber">(f)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-7-1-1">Suppose <span class="process-math">\(\L(\theta)\)</span> is a likelihood function, and <span class="process-math">\(\L(2) = 0.04\text{.}\)</span> Then the probability that <span class="process-math">\(\theta = 2\)</span> is <span class="process-math">\(0.04\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-7-2" 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="exam-01-3-1-7-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-7-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.f.1"><a href="#exam-01-3-1-7-2" title="Copy heading and permalink for Solution 1.f.1" aria-label="Copy heading and permalink for Solution 1.f.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.f"><a href="#exam-01-3-1-7" title="Copy heading and permalink for Task 1.f" aria-label="Copy heading and permalink for Task 1.f">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#exam-01-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><section class="onepage" id="exam-01-4"><article class="exercise exercise-like" id="exam-01-4-1"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-4-1-1">
|
||||
<div class="para" id="exam-01-4-1-1-1">Consider the sample space <span class="process-math">\(S = \{1, 2, 3, 4, 5\}\)</span> with probability distribution given in the table below. Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{1, 3, 5\}\text{.}\)</span> Calculate the following probabilities.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<article class="task exercise-like" id="exam-01-4-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-2-1-1">
|
||||
<span class="process-math">\(\Pr(A \cup B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-2-2" 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" id="exam-01-4-1-2-2-1">
|
||||
<span class="process-math">\(A\cup B = \{ 1, 2, 3, 5 \}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cup B) = 0.1 + 0.2 + 0.3 + 0.1 = \boxed{0.7}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.a.1"><a href="#exam-01-4-1-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 2.a"><a href="#exam-01-4-1-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-4-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-3-1-1">
|
||||
<span class="process-math">\(\Pr(A \cap B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-3-2" 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="exam-01-4-1-3-2-1">
|
||||
<span class="process-math">\(A\cap B = \{ 1, 3 \}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = 0.1 + 0.3 = \boxed{0.4}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.b.1"><a href="#exam-01-4-1-3-2" 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>
|
||||
<div class="autopermalink" data-description="Task 2.b"><a href="#exam-01-4-1-3" title="Copy heading and permalink for Task 2.b" aria-label="Copy heading and permalink for Task 2.b">🔗</a></div></article><article class="task exercise-like" id="exam-01-4-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-4-1-1">
|
||||
<span class="process-math">\(\Pr(A \mid B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-4-2" 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="exam-01-4-1-4-2-1">
|
||||
<span class="process-math">\(\Pr(A\mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{0.4}{0.1 + 0.3 + 0.1} = \frac{0.4}{0.5} = \boxed{0.8}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.c.1"><a href="#exam-01-4-1-4-2" title="Copy heading and permalink for Solution 2.c.1" aria-label="Copy heading and permalink for Solution 2.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 2.c"><a href="#exam-01-4-1-4" title="Copy heading and permalink for Task 2.c" aria-label="Copy heading and permalink for Task 2.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#exam-01-4-1" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><section class="onepage" id="exam-01-5"><article class="exercise exercise-like" id="exam-01-5-1"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-5-1-1">
|
||||
<div class="para" id="exam-01-5-1-1-1">Consider the random variable <span class="process-math">\(X\)</span> with probability distribution below.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<article class="task exercise-like" id="exam-01-5-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-5-1-2-1-1">Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-1-2-2" 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-01-5-1-2-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-5-1-2-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X) \amp = (1)(0.2) + (2)(0.3) + (3)(0.1) + (4)(0.25) + (5)(0.15) = \boxed{2.85}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 3.a.1"><a href="#exam-01-5-1-2-2" title="Copy heading and permalink for Solution 3.a.1" aria-label="Copy heading and permalink for Solution 3.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="1in"></div>
|
||||
<div class="autopermalink" data-description="Task 3.a"><a href="#exam-01-5-1-2" title="Copy heading and permalink for Task 3.a" aria-label="Copy heading and permalink for Task 3.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-5-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-5-1-3-1-1">Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-1-3-2" 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 logical" id="exam-01-5-1-3-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-5-1-3-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X^2) \amp = (1)^2(0.2) + (2)^2(0.3) + (3)^2(0.1) + (4)^2(0.25) + (5)^2(0.15) = 10.05 \\
|
||||
\Var(X) \amp = \E(X^2) - \left(\E(X)\right)^2 = 10.05 - (2.85)^2 = \boxed{1.9275}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 3.b.1"><a href="#exam-01-5-1-3-2" title="Copy heading and permalink for Solution 3.b.1" aria-label="Copy heading and permalink for Solution 3.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="1in"></div>
|
||||
<div class="autopermalink" data-description="Task 3.b"><a href="#exam-01-5-1-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#exam-01-5-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-5-2"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-5-2-1-1">A diagnostic test is developed to detect a disease present in 2.1% of the population. For a patient who has the disease, the test will accurately give a positive result 76% of the time. When the patient does not have the disease, the test will accurately give a negative result 98.2% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="exam-01-5-2-1-2">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-01-5-2-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-2-2" 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-01-5-2-2-1">
|
||||
<div class="para">Let <span class="process-math">\(P\)</span> be the event of receiving a positive test result and <span class="process-math">\(D\)</span> be the event of having the disease. The given information is: <span class="process-math">\(\Pr(D) = 0.021, \Pr(P\mid D) = 0.76\text{,}\)</span> and <span class="process-math">\(\Pr(P^c \mid D^c) = 0.982\text{.}\)</span> Then, using Bayes’ Theorem:</div>
|
||||
<div class="displaymath process-math" id="exam-01-5-2-2-1-5">
|
||||
\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) + \Pr(P^c \mid D)\Pr(D)} \\
|
||||
\amp = \frac{\Pr(P^c \mid D^c)(1 - \Pr(D))}{\Pr(P^c \mid D^c)(1 - \Pr(D)) + (1 - \Pr(P \mid D))\Pr(D)} \\
|
||||
\amp = \frac{(0.982)(1 - 0.021)}{(0.982)(1 - 0.021) + (1 - 0.76)(0.021)} \\
|
||||
\amp \approx \boxed{0.995}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 4.1"><a href="#exam-01-5-2-2" title="Copy heading and permalink for Solution 4.1" aria-label="Copy heading and permalink for Solution 4.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="2in"></div>
|
||||
<div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#exam-01-5-2" 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="exam-01-6"><article class="exercise exercise-like" id="exam-01-6-1"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-6-1-1"><div class="para" id="exam-01-6-1-1-1">In each of the following scenarios, state the maximum likelihood estimation for the unknown parameter indicated.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-1-1" 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-01-6-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-6-1-2-1-1">A radioactive material is observed for 5 hours. 120 particle emissions are seen. <span class="process-math">\(\lambda\)</span> is the hourly rate of particle emissions.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-1-2-2" 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" id="exam-01-6-1-2-2-1">120 emissions per 5 hours is an hourly rate of <span class="process-math">\(\boxed{\widehat{\lambda} = 120/5 = 24 \text{ per hour}}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 5.a.1"><a href="#exam-01-6-1-2-2" title="Copy heading and permalink for Solution 5.a.1" aria-label="Copy heading and permalink for Solution 5.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 5.a"><a href="#exam-01-6-1-2" title="Copy heading and permalink for Task 5.a" aria-label="Copy heading and permalink for Task 5.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-6-1-3-1-1">In each of 5 trials, a coin is flipped until heads is seen. The number of flips in each trial is 3, 4, 3, 5, and 6. <span class="process-math">\(p\)</span> is the probability of the coin coming up heads on a flip.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-1-3-2" 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="exam-01-6-1-3-2-1">There are <span class="process-math">\(3 + 4 + 3 + 5 + 6 = 21\)</span> total flips, <span class="process-math">\(5\)</span> of which are heads, so the MLE is <span class="process-math">\(\boxed{\widehat{p} = \frac{5}{21}}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 5.b.1"><a href="#exam-01-6-1-3-2" title="Copy heading and permalink for Solution 5.b.1" aria-label="Copy heading and permalink for Solution 5.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 5.b"><a href="#exam-01-6-1-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#exam-01-6-1" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-6-2"><h3 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-6-2-1"><div class="para" id="exam-01-6-2-1-1">Let <span class="process-math">\(X\)</span> be a continuous random variable with c.d.f. <span class="process-math">\(\displaystyle{F(x) = \frac{1}{8}(x^3 + 3x + 4)}\text{,}\)</span> where <span class="process-math">\(x \in [-1, 1]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-1-1" 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-01-6-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-2-1-1">Find the p.d.f. <span class="process-math">\(f(x)\)</span> for <span class="process-math">\(X\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-2-2-2" 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" id="exam-01-6-2-2-2-1">
|
||||
<span class="process-math">\(f(x) = F'(x) = \frac{1}{8}\left(3x^2 + 3\right) = \frac{3}{8}\left(x^2 + 1\right)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 6.a.1"><a href="#exam-01-6-2-2-2" title="Copy heading and permalink for Solution 6.a.1" aria-label="Copy heading and permalink for Solution 6.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 6.a"><a href="#exam-01-6-2-2" title="Copy heading and permalink for Task 6.a" aria-label="Copy heading and permalink for Task 6.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-3-1-1">Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-2-3-2" 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 logical" id="exam-01-6-2-3-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-6-2-3-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X) \amp = \int_{-1}^1 x \cdot f(x)\ dx = \int_{-1}^1 x\cdot \frac{3}{8}\left(x^2 + 1\right)\ dx \\
|
||||
\amp = \frac{3}{8} \int_{-1}^1 x^3 + x\ dx = \frac{3}{8} \left(\frac{x^4}{4} + \frac{x^2}{2}\right)\bigg|_{-1}^1 \\
|
||||
\amp = \frac{3}{8} \left[\left(\frac{1}{4} + \frac{1}{2}\right) - \left( \frac{1}{4} + \frac{1}{2}\right)\right] = \boxed{0}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 6.b.1"><a href="#exam-01-6-2-3-2" title="Copy heading and permalink for Solution 6.b.1" aria-label="Copy heading and permalink for Solution 6.b.1">🔗</a></div>
|
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</div></details></div>
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<div class="autopermalink" data-description="Task 6.b"><a href="#exam-01-6-2-3" title="Copy heading and permalink for Task 6.b" aria-label="Copy heading and permalink for Task 6.b">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-2-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-4-1-1">Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-01-6-2-4-2" 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-01-6-2-4-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-6-2-4-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X^2) \amp = \int_{-1}^1 x^2 \cdot f(x)\ dx = \int_{-1}^1 x^2\cdot \frac{3}{8}\left(x^2 + 1\right)\ dx \\
|
||||
\amp = \frac{3}{8} \int_{-1}^1 x^4 + x^2\ dx = \frac{3}{8} \left(\frac{x^5}{5} + \frac{x^3}{3}\right)\bigg|_{-1}^1 \\
|
||||
\amp = \frac{3}{8} \left[\left(\frac{1}{5} + \frac{1}{3}\right) - \left( \frac{-1}{5} - \frac{1}{3}\right)\right] = \frac{3}{8} \cdot \frac{16}{15} = \frac{2}{5} \\
|
||||
\Var(X) \amp = \E(X^2) - (\E(X))^2 = \frac{2}{5} - 0^2 = \boxed{\frac{2}{5}}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
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<div class="autopermalink" data-description="Solution 6.c.1"><a href="#exam-01-6-2-4-2" title="Copy heading and permalink for Solution 6.c.1" aria-label="Copy heading and permalink for Solution 6.c.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Task 6.c"><a href="#exam-01-6-2-4" title="Copy heading and permalink for Task 6.c" aria-label="Copy heading and permalink for Task 6.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 6"><a href="#exam-01-6-2" title="Copy heading and permalink for Worksheet Exercise 6" aria-label="Copy heading and permalink for Worksheet Exercise 6">🔗</a></div></article></section><section class="onepage" id="exam-01-7"><article class="exercise exercise-like" id="exam-01-7-1"><h3 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-7-1-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below. Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <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">\(Y = 1\)</span></td>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 2\)</span></td>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
<div class="solutions"><details id="exam-01-7-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="exam-01-7-1-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-7-1-2-1-1">
|
||||
\begin{align*}
|
||||
\Pr(X = 0, Y = 1) \amp = 0.1 \\
|
||||
\Pr(X = 0) \amp = 0.1 + 0.2 = 0.3 \\
|
||||
\Pr(Y = 1) \amp = 0.1 + 0.25 + 0.2 = 0.55 \\
|
||||
\Pr(X = 0)\Pr(Y = 1) \amp = (0.3)(0.55) = 0.165 \neq 0.1,
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">so <span class="process-math">\(X, Y\)</span> are not independent.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 7.1"><a href="#exam-01-7-1-2" title="Copy heading and permalink for Solution 7.1" aria-label="Copy heading and permalink for Solution 7.1">🔗</a></div>
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</div></details></div>
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<div class="workspace" data-space="1in"></div>
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<div class="autopermalink" data-description="Worksheet Exercise 7"><a href="#exam-01-7-1" title="Copy heading and permalink for Worksheet Exercise 7" aria-label="Copy heading and permalink for Worksheet Exercise 7">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-7-2"><h3 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-7-2-1-1">Suppose a parameter <span class="process-math">\(-1/2 \leq \theta \leq 1\)</span> has likelihood function <span class="process-math">\(\L(\theta) = \theta^2 - \theta^3\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exam-01-7-2-2" 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">
|
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<div class="para logical" id="exam-01-7-2-2-1">
|
||||
<div class="para">
|
||||
<span class="process-math">\(\L\)</span> is a continuous function and <span class="process-math">\([-1/2, 1]\)</span> is a closed interval, so we use the CIM.</div>
|
||||
<div class="displaymath process-math" id="exam-01-7-2-2-1-3">
|
||||
\begin{align*}
|
||||
\L'(\theta) \amp = 2\theta - 3\theta^2 \\
|
||||
\L'(\theta) = 0 \text{ when } 0 \amp = 2\theta - 3\theta^2 \\
|
||||
0 \amp =\theta (2 - 3\theta) \\
|
||||
\theta \amp = 0, \frac{2}{3}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Then:</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(\theta\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(-1/2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2/3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\L(\theta)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/8\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/27\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
<div class="para" id="exam-01-7-2-2-3">
|
||||
<span class="process-math">\(3/8\)</span> is the largest value, so the MLE is <span class="process-math">\(\boxed{\widehat{\theta} = -1/2}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-2-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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</div></details></div>
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<div class="workspace" data-space="2in"></div>
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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>
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<p class="byline">Andy Eisenberg</p>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="worksheet" id="exam-01" data-margins="0.75in 0.75in 0.75in 0.75in"><h2 class="heading hide-type">
|
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<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">Exam 1</span><div class="print-links"><a href="exam-01-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></div>
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</h2>
|
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<section class="introduction" id="exam-01-2"><div class="para" id="exam-01-2-1">Show all relevant work.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></section><section class="onepage" id="exam-01-3"><article class="exercise exercise-like" id="exam-01-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-3-1-1"><div class="para" id="exam-01-3-1-1-1">Write either True or False for each of the following statements. No justification is required.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-1-1" 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-01-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-2-1-1">Let <span class="process-math">\(A, B\)</span> be any sets. Then <span class="process-math">\(|A \cap B| = |A| \cdot |B|\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-2-2" 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" id="exam-01-3-1-2-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.a.1"><a href="#exam-01-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.a"><a href="#exam-01-3-1-2" 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-01-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-3-1-1">Let <span class="process-math">\(A, B\)</span> be any events. Then <span class="process-math">\(\Pr(A \mid B) = \Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-3-2" 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="exam-01-3-1-3-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.b.1"><a href="#exam-01-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.b"><a href="#exam-01-3-1-3" 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-01-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-4-1-1">Given a joint distribution for random variables <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{,}\)</span> then it must be the case that <span class="process-math">\(\Pr(X = 0, Y = 0) = \Pr(X = 0)\Pr(Y = 0)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-4-2" 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="exam-01-3-1-4-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.c.1"><a href="#exam-01-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.c"><a href="#exam-01-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-5"><h4 class="heading"><span class="codenumber">(d)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-5-1-1">Suppose we flip a coin repeatedly until we first see heads and let <span class="process-math">\(T\)</span> be the number of flips. Then <span class="process-math">\(\Pr(T = 2) \geq \Pr(T = 4)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-5-2" 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="exam-01-3-1-5-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.d.1"><a href="#exam-01-3-1-5-2" title="Copy heading and permalink for Solution 1.d.1" aria-label="Copy heading and permalink for Solution 1.d.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.d"><a href="#exam-01-3-1-5" title="Copy heading and permalink for Task 1.d" aria-label="Copy heading and permalink for Task 1.d">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-6"><h4 class="heading"><span class="codenumber">(e)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-6-1-1">If molecules are observed leaving a cell after 1.5 minutes, 1.8 minutes, 2.1 minutes, 2.2 minutes, and 2.4 minutes, then the maximum likelihood estimation for the rate at which molecules leave the cell is <span class="process-math">\(1/2\)</span> per minute.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-6-2" 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="exam-01-3-1-6-2-1">True.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-6-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.e.1"><a href="#exam-01-3-1-6-2" title="Copy heading and permalink for Solution 1.e.1" aria-label="Copy heading and permalink for Solution 1.e.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.e"><a href="#exam-01-3-1-6" title="Copy heading and permalink for Task 1.e" aria-label="Copy heading and permalink for Task 1.e">🔗</a></div></article><article class="task exercise-like" id="exam-01-3-1-7"><h4 class="heading"><span class="codenumber">(f)</span></h4>
|
||||
<div class="para" id="exam-01-3-1-7-1-1">Suppose <span class="process-math">\(\L(\theta)\)</span> is a likelihood function, and <span class="process-math">\(\L(2) = 0.04\text{.}\)</span> Then the probability that <span class="process-math">\(\theta = 2\)</span> is <span class="process-math">\(0.04\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-3-1-7-2" 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="exam-01-3-1-7-2-1">False.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-3-1-7-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 1.f.1"><a href="#exam-01-3-1-7-2" title="Copy heading and permalink for Solution 1.f.1" aria-label="Copy heading and permalink for Solution 1.f.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 1.f"><a href="#exam-01-3-1-7" title="Copy heading and permalink for Task 1.f" aria-label="Copy heading and permalink for Task 1.f">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#exam-01-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><section class="onepage" id="exam-01-4"><article class="exercise exercise-like" id="exam-01-4-1"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-4-1-1">
|
||||
<div class="para" id="exam-01-4-1-1-1">Consider the sample space <span class="process-math">\(S = \{1, 2, 3, 4, 5\}\)</span> with probability distribution given in the table below. Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{1, 3, 5\}\text{.}\)</span> Calculate the following probabilities.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<article class="task exercise-like" id="exam-01-4-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-2-1-1">
|
||||
<span class="process-math">\(\Pr(A \cup B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-2-2" 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" id="exam-01-4-1-2-2-1">
|
||||
<span class="process-math">\(A\cup B = \{ 1, 2, 3, 5 \}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cup B) = 0.1 + 0.2 + 0.3 + 0.1 = \boxed{0.7}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.a.1"><a href="#exam-01-4-1-2-2" title="Copy heading and permalink for Solution 2.a.1" aria-label="Copy heading and permalink for Solution 2.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 2.a"><a href="#exam-01-4-1-2" title="Copy heading and permalink for Task 2.a" aria-label="Copy heading and permalink for Task 2.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-4-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-3-1-1">
|
||||
<span class="process-math">\(\Pr(A \cap B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-3-2" 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="exam-01-4-1-3-2-1">
|
||||
<span class="process-math">\(A\cap B = \{ 1, 3 \}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = 0.1 + 0.3 = \boxed{0.4}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.b.1"><a href="#exam-01-4-1-3-2" 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>
|
||||
<div class="autopermalink" data-description="Task 2.b"><a href="#exam-01-4-1-3" title="Copy heading and permalink for Task 2.b" aria-label="Copy heading and permalink for Task 2.b">🔗</a></div></article><article class="task exercise-like" id="exam-01-4-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-4-1-4-1-1">
|
||||
<span class="process-math">\(\Pr(A \mid B)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-4-1-4-2" 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="exam-01-4-1-4-2-1">
|
||||
<span class="process-math">\(\Pr(A\mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{0.4}{0.1 + 0.3 + 0.1} = \frac{0.4}{0.5} = \boxed{0.8}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-4-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 2.c.1"><a href="#exam-01-4-1-4-2" title="Copy heading and permalink for Solution 2.c.1" aria-label="Copy heading and permalink for Solution 2.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 2.c"><a href="#exam-01-4-1-4" title="Copy heading and permalink for Task 2.c" aria-label="Copy heading and permalink for Task 2.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#exam-01-4-1" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article></section><section class="onepage" id="exam-01-5"><article class="exercise exercise-like" id="exam-01-5-1"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-5-1-1">
|
||||
<div class="para" id="exam-01-5-1-1-1">Consider the random variable <span class="process-math">\(X\)</span> with probability distribution below.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<article class="task exercise-like" id="exam-01-5-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-5-1-2-1-1">Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-1-2-2" 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-01-5-1-2-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-5-1-2-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X) \amp = (1)(0.2) + (2)(0.3) + (3)(0.1) + (4)(0.25) + (5)(0.15) = \boxed{2.85}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 3.a.1"><a href="#exam-01-5-1-2-2" title="Copy heading and permalink for Solution 3.a.1" aria-label="Copy heading and permalink for Solution 3.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="1in"></div>
|
||||
<div class="autopermalink" data-description="Task 3.a"><a href="#exam-01-5-1-2" title="Copy heading and permalink for Task 3.a" aria-label="Copy heading and permalink for Task 3.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-5-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-5-1-3-1-1">Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-1-3-2" 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 logical" id="exam-01-5-1-3-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-5-1-3-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X^2) \amp = (1)^2(0.2) + (2)^2(0.3) + (3)^2(0.1) + (4)^2(0.25) + (5)^2(0.15) = 10.05 \\
|
||||
\Var(X) \amp = \E(X^2) - \left(\E(X)\right)^2 = 10.05 - (2.85)^2 = \boxed{1.9275}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 3.b.1"><a href="#exam-01-5-1-3-2" title="Copy heading and permalink for Solution 3.b.1" aria-label="Copy heading and permalink for Solution 3.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="1in"></div>
|
||||
<div class="autopermalink" data-description="Task 3.b"><a href="#exam-01-5-1-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#exam-01-5-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-5-2"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-5-2-1-1">A diagnostic test is developed to detect a disease present in 2.1% of the population. For a patient who has the disease, the test will accurately give a positive result 76% of the time. When the patient does not have the disease, the test will accurately give a negative result 98.2% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="exam-01-5-2-1-2">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-01-5-2-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-5-2-2" 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-01-5-2-2-1">
|
||||
<div class="para">Let <span class="process-math">\(P\)</span> be the event of receiving a positive test result and <span class="process-math">\(D\)</span> be the event of having the disease. The given information is: <span class="process-math">\(\Pr(D) = 0.021, \Pr(P\mid D) = 0.76\text{,}\)</span> and <span class="process-math">\(\Pr(P^c \mid D^c) = 0.982\text{.}\)</span> Then, using Bayes’ Theorem:</div>
|
||||
<div class="displaymath process-math" id="exam-01-5-2-2-1-5">
|
||||
\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) + \Pr(P^c \mid D)\Pr(D)} \\
|
||||
\amp = \frac{\Pr(P^c \mid D^c)(1 - \Pr(D))}{\Pr(P^c \mid D^c)(1 - \Pr(D)) + (1 - \Pr(P \mid D))\Pr(D)} \\
|
||||
\amp = \frac{(0.982)(1 - 0.021)}{(0.982)(1 - 0.021) + (1 - 0.76)(0.021)} \\
|
||||
\amp \approx \boxed{0.995}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-5-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 4.1"><a href="#exam-01-5-2-2" title="Copy heading and permalink for Solution 4.1" aria-label="Copy heading and permalink for Solution 4.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="2in"></div>
|
||||
<div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#exam-01-5-2" 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="exam-01-6"><article class="exercise exercise-like" id="exam-01-6-1"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-6-1-1"><div class="para" id="exam-01-6-1-1-1">In each of the following scenarios, state the maximum likelihood estimation for the unknown parameter indicated.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-1-1" 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-01-6-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-6-1-2-1-1">A radioactive material is observed for 5 hours. 120 particle emissions are seen. <span class="process-math">\(\lambda\)</span> is the hourly rate of particle emissions.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-1-2-2" 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" id="exam-01-6-1-2-2-1">120 emissions per 5 hours is an hourly rate of <span class="process-math">\(\boxed{\widehat{\lambda} = 120/5 = 24 \text{ per hour}}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 5.a.1"><a href="#exam-01-6-1-2-2" title="Copy heading and permalink for Solution 5.a.1" aria-label="Copy heading and permalink for Solution 5.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 5.a"><a href="#exam-01-6-1-2" title="Copy heading and permalink for Task 5.a" aria-label="Copy heading and permalink for Task 5.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-6-1-3-1-1">In each of 5 trials, a coin is flipped until heads is seen. The number of flips in each trial is 3, 4, 3, 5, and 6. <span class="process-math">\(p\)</span> is the probability of the coin coming up heads on a flip.<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-1-3-2" 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="exam-01-6-1-3-2-1">There are <span class="process-math">\(3 + 4 + 3 + 5 + 6 = 21\)</span> total flips, <span class="process-math">\(5\)</span> of which are heads, so the MLE is <span class="process-math">\(\boxed{\widehat{p} = \frac{5}{21}}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 5.b.1"><a href="#exam-01-6-1-3-2" title="Copy heading and permalink for Solution 5.b.1" aria-label="Copy heading and permalink for Solution 5.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 5.b"><a href="#exam-01-6-1-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#exam-01-6-1" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-6-2"><h3 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exam-01-6-2-1"><div class="para" id="exam-01-6-2-1-1">Let <span class="process-math">\(X\)</span> be a continuous random variable with c.d.f. <span class="process-math">\(\displaystyle{F(x) = \frac{1}{8}(x^3 + 3x + 4)}\text{,}\)</span> where <span class="process-math">\(x \in [-1, 1]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-1-1" 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-01-6-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-2-1-1">Find the p.d.f. <span class="process-math">\(f(x)\)</span> for <span class="process-math">\(X\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-2-2-2" 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" id="exam-01-6-2-2-2-1">
|
||||
<span class="process-math">\(f(x) = F'(x) = \frac{1}{8}\left(3x^2 + 3\right) = \frac{3}{8}\left(x^2 + 1\right)\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 6.a.1"><a href="#exam-01-6-2-2-2" title="Copy heading and permalink for Solution 6.a.1" aria-label="Copy heading and permalink for Solution 6.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 6.a"><a href="#exam-01-6-2-2" title="Copy heading and permalink for Task 6.a" aria-label="Copy heading and permalink for Task 6.a">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-3-1-1">Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-2-3-2" 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 logical" id="exam-01-6-2-3-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-6-2-3-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X) \amp = \int_{-1}^1 x \cdot f(x)\ dx = \int_{-1}^1 x\cdot \frac{3}{8}\left(x^2 + 1\right)\ dx \\
|
||||
\amp = \frac{3}{8} \int_{-1}^1 x^3 + x\ dx = \frac{3}{8} \left(\frac{x^4}{4} + \frac{x^2}{2}\right)\bigg|_{-1}^1 \\
|
||||
\amp = \frac{3}{8} \left[\left(\frac{1}{4} + \frac{1}{2}\right) - \left( \frac{1}{4} + \frac{1}{2}\right)\right] = \boxed{0}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 6.b.1"><a href="#exam-01-6-2-3-2" title="Copy heading and permalink for Solution 6.b.1" aria-label="Copy heading and permalink for Solution 6.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 6.b"><a href="#exam-01-6-2-3" title="Copy heading and permalink for Task 6.b" aria-label="Copy heading and permalink for Task 6.b">🔗</a></div></article><article class="task exercise-like" id="exam-01-6-2-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exam-01-6-2-4-1-1">Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-6-2-4-2" 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-01-6-2-4-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-6-2-4-2-1-1">
|
||||
\begin{align*}
|
||||
\E(X^2) \amp = \int_{-1}^1 x^2 \cdot f(x)\ dx = \int_{-1}^1 x^2\cdot \frac{3}{8}\left(x^2 + 1\right)\ dx \\
|
||||
\amp = \frac{3}{8} \int_{-1}^1 x^4 + x^2\ dx = \frac{3}{8} \left(\frac{x^5}{5} + \frac{x^3}{3}\right)\bigg|_{-1}^1 \\
|
||||
\amp = \frac{3}{8} \left[\left(\frac{1}{5} + \frac{1}{3}\right) - \left( \frac{-1}{5} - \frac{1}{3}\right)\right] = \frac{3}{8} \cdot \frac{16}{15} = \frac{2}{5} \\
|
||||
\Var(X) \amp = \E(X^2) - (\E(X))^2 = \frac{2}{5} - 0^2 = \boxed{\frac{2}{5}}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-6-2-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 6.c.1"><a href="#exam-01-6-2-4-2" title="Copy heading and permalink for Solution 6.c.1" aria-label="Copy heading and permalink for Solution 6.c.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 6.c"><a href="#exam-01-6-2-4" title="Copy heading and permalink for Task 6.c" aria-label="Copy heading and permalink for Task 6.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 6"><a href="#exam-01-6-2" title="Copy heading and permalink for Worksheet Exercise 6" aria-label="Copy heading and permalink for Worksheet Exercise 6">🔗</a></div></article></section><section class="onepage" id="exam-01-7"><article class="exercise exercise-like" id="exam-01-7-1"><h3 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-7-1-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below. Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <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">\(Y = 1\)</span></td>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 2\)</span></td>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
<div class="solutions"><details id="exam-01-7-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||||
<div class="para logical" id="exam-01-7-1-2-1">
|
||||
<div class="displaymath process-math" id="exam-01-7-1-2-1-1">
|
||||
\begin{align*}
|
||||
\Pr(X = 0, Y = 1) \amp = 0.1 \\
|
||||
\Pr(X = 0) \amp = 0.1 + 0.2 = 0.3 \\
|
||||
\Pr(Y = 1) \amp = 0.1 + 0.25 + 0.2 = 0.55 \\
|
||||
\Pr(X = 0)\Pr(Y = 1) \amp = (0.3)(0.55) = 0.165 \neq 0.1,
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">so <span class="process-math">\(X, Y\)</span> are not independent.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 7.1"><a href="#exam-01-7-1-2" title="Copy heading and permalink for Solution 7.1" aria-label="Copy heading and permalink for Solution 7.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="1in"></div>
|
||||
<div class="autopermalink" data-description="Worksheet Exercise 7"><a href="#exam-01-7-1" title="Copy heading and permalink for Worksheet Exercise 7" aria-label="Copy heading and permalink for Worksheet Exercise 7">🔗</a></div></article><article class="exercise exercise-like" id="exam-01-7-2"><h3 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exam-01-7-2-1-1">Suppose a parameter <span class="process-math">\(-1/2 \leq \theta \leq 1\)</span> has likelihood function <span class="process-math">\(\L(\theta) = \theta^2 - \theta^3\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="exam-01-7-2-2" 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-01-7-2-2-1">
|
||||
<div class="para">
|
||||
<span class="process-math">\(\L\)</span> is a continuous function and <span class="process-math">\([-1/2, 1]\)</span> is a closed interval, so we use the CIM.</div>
|
||||
<div class="displaymath process-math" id="exam-01-7-2-2-1-3">
|
||||
\begin{align*}
|
||||
\L'(\theta) \amp = 2\theta - 3\theta^2 \\
|
||||
\L'(\theta) = 0 \text{ when } 0 \amp = 2\theta - 3\theta^2 \\
|
||||
0 \amp =\theta (2 - 3\theta) \\
|
||||
\theta \amp = 0, \frac{2}{3}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Then:</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(\theta\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(-1/2\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2/3\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\L(\theta)\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/8\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/27\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
</tr>
|
||||
</table></div>
|
||||
<div class="para" id="exam-01-7-2-2-3">
|
||||
<span class="process-math">\(3/8\)</span> is the largest value, so the MLE is <span class="process-math">\(\boxed{\widehat{\theta} = -1/2}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exam-01-7-2-2-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Solution 8.1"><a href="#exam-01-7-2-2" title="Copy heading and permalink for Solution 8.1" aria-label="Copy heading and permalink for Solution 8.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="workspace" data-space="2in"></div>
|
||||
<div class="autopermalink" data-description="Worksheet Exercise 8"><a href="#exam-01-7-2" title="Copy heading and permalink for Worksheet Exercise 8" aria-label="Copy heading and permalink for Worksheet Exercise 8">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Exam 1"><a href="#exam-01" title="Copy heading and permalink for Worksheet: Exam 1" aria-label="Copy heading and permalink for Worksheet: Exam 1">🔗</a></div></section></div>
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||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
|
||||
@@ -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"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="quizzes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="recitations.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="quizzes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="exams.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</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}
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
@@ -404,7 +408,7 @@ f(x) = F'(x) = 4x - 4x^3.
|
||||
<div class="autopermalink" data-description="Task 3.b"><a href="#quiz-02-4-1-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#quiz-02-4-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 2"><a href="#quiz-02" title="Copy heading and permalink for Worksheet: Quiz 2" aria-label="Copy heading and permalink for Worksheet: Quiz 2">🔗</a></div></section>
|
||||
</div>
|
||||
<div id="ptx-content-footer" class="ptx-content-footer">
|
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<a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitations.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
|
||||
<a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="exams.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
|
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</div></main>
|
||||
</div>
|
||||
<div id="ptx-page-footer" class="ptx-page-footer">
|
||||
|
||||
+6
-2
@@ -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"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="quizzes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="recitations.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="quizzes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="exams.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</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}
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
@@ -388,7 +392,7 @@ f(x) = F'(x) = 4x - 4x^3.
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 3.b"><a href="#quiz-02-4-1-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#quiz-02-4-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 2"><a href="#quiz-02" title="Copy heading and permalink for Worksheet: Quiz 2" aria-label="Copy heading and permalink for Worksheet: Quiz 2">🔗</a></div></section></div>
|
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<div id="ptx-content-footer" class="ptx-content-footer">
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<a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitations.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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<a class="previous-button button" href="quiz-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="exams.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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</div></main>
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</div>
|
||||
<div id="ptx-page-footer" class="ptx-page-footer">
|
||||
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
|
||||
@@ -240,6 +240,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list contains-active"><li class="toc-item toc-worksheet active"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
|
||||
|
||||
@@ -240,6 +240,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list contains-active"><li class="toc-item toc-worksheet active"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
|
||||
|
||||
@@ -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"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="quiz-02.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="exam-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</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}
|
||||
@@ -240,6 +240,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list active"><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>
|
||||
@@ -260,7 +264,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</h2>
|
||||
<nav class="summary-links"><ul><li><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></li></ul></nav><div class="autopermalink" data-description="Chapter: Recitations"><a href="#recitations" title="Copy heading and permalink for Chapter: Recitations" aria-label="Copy heading and permalink for Chapter: Recitations">🔗</a></div></section></div>
|
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<div id="ptx-content-footer" class="ptx-content-footer">
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<a class="previous-button button" href="quiz-02.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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<a class="previous-button button" href="exam-01.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="recitation-calculus-review.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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</div></main>
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</div>
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<div id="ptx-page-footer" class="ptx-page-footer">
|
||||
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
</li>
|
||||
|
||||
@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
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<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
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<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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@@ -260,6 +264,7 @@ eBookConfig.enable_chatcodes = false;
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<li><a href="course-documents.html" class="internal"><span class="title">Course Documents</span></a></li>
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<li><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></li>
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<li><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></li>
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<li><a href="exams.html" class="internal"><span class="title">Exams</span></a></li>
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<li><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></li>
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<li><a href="review.html" class="internal"><span class="title">Exam Review</span></a></li>
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</ul></nav><div class="autopermalink" data-description="Book: Introduction to Probability and Statistics for the Life Sciences"><a href="#root-1-2" title="Copy heading and permalink for Book: Introduction to Probability and Statistics for the Life Sciences" aria-label="Copy heading and permalink for Book: Introduction to Probability and Statistics for the Life Sciences">🔗</a></div></section></div>
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@@ -241,6 +241,10 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-chapter">
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<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</span></a></div></li></ul>
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<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>
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<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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