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@@ -2,7 +2,7 @@
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<html>
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<!--********************************************-->
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<!--* Generated from PreTeXt source *-->
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<!--* on 2026-02-05T07:22:54-05:00 *-->
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<!--* on 2026-02-05T10:22:20-05:00 *-->
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<!--* A recent stable commit (2022-07-01): *-->
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<!--* 6c761d3dba23af92cba35001c852aac04ae99a5f *-->
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<!--* *-->
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@@ -808,7 +808,7 @@ var ptx_lunr_docs = [
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"type": "Worksheet",
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"number": "",
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"title": "Quiz 2",
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"body": " Quiz 2 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Since , and are not independent. Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . "
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"body": " Quiz 2 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? , and . Since , and are not independent. Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . "
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},
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{
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"id": "quiz-02-3-1",
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@@ -826,7 +826,7 @@ var ptx_lunr_docs = [
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"type": "Worksheet Exercise",
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"number": "2",
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"title": "",
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"body": " Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Since , and are not independent. "
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"body": " Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? , and . Since , and are not independent. "
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},
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{
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"id": "quiz-02-4-1",
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@@ -309,11 +309,9 @@ eBookConfig.enable_chatcodes = false;
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<div class="para logical" id="quiz-02-3-2-2-2-1">
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<div class="displaymath process-math" id="quiz-02-3-2-2-2-1-1">
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\begin{align*}
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\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\
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\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \\
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\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 \\
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\Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\
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\Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75
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\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \amp \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\
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\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \amp \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 \\
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\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4
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\end{align*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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@@ -324,15 +322,8 @@ eBookConfig.enable_chatcodes = false;
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<div class="para" id="quiz-02-3-2-3-1-1">Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="quiz-02-3-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para logical" id="quiz-02-3-2-3-2-1">
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<div class="displaymath process-math" id="quiz-02-3-2-3-2-1-1">
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\begin{align*}
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\Pr(X = 2, Y = 0) \amp = 0.2 \\
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\Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1
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\end{align*}
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</div>
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<div class="para">Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="quiz-02-3-2-3-2-1">
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<span class="process-math">\(\Pr(X = 2, Y = 0) = 0.2 \text{,}\)</span> and <span class="process-math">\(\Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1\text{.}\)</span> Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-02-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
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</div></details></div>
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+5
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@@ -294,11 +294,9 @@ eBookConfig.enable_chatcodes = false;
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<div class="para logical" id="quiz-02-3-2-2-2-1">
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<div class="displaymath process-math" id="quiz-02-3-2-2-2-1-1">
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\begin{align*}
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\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\
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\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \\
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\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 \\
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\Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\
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\Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75
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\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \amp \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\
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\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \amp \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 \\
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\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4
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\end{align*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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@@ -309,15 +307,8 @@ eBookConfig.enable_chatcodes = false;
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<div class="para" id="quiz-02-3-2-3-1-1">Are <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="quiz-02-3-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
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<div class="para logical" id="quiz-02-3-2-3-2-1">
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<div class="displaymath process-math" id="quiz-02-3-2-3-2-1-1">
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\begin{align*}
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\Pr(X = 2, Y = 0) \amp = 0.2 \\
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\Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1
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\end{align*}
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</div>
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<div class="para">Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="quiz-02-3-2-3-2-1">
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<span class="process-math">\(\Pr(X = 2, Y = 0) = 0.2 \text{,}\)</span> and <span class="process-math">\(\Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1\text{.}\)</span> Since <span class="process-math">\(0.2 \neq 0.1\text{,}\)</span> <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Solution 2.b.1"><a href="#quiz-02-3-2-3-2" title="Copy heading and permalink for Solution 2.b.1" aria-label="Copy heading and permalink for Solution 2.b.1">🔗</a></div>
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</div></details></div>
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