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@@ -531,21 +531,54 @@ f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigm
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.5</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercises-Continuous-RVs-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Continuous-RVs-1-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span> What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="introduction" id="exercises-Continuous-RVs-1-1"><div class="para" id="exercises-Continuous-RVs-1-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-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="exercises-Continuous-RVs-1-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Continuous-RVs-1-2-1-1">What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 2.2.5.1"><a href="#exercises-Continuous-RVs-1" title="Copy heading and permalink for Exercise 2.2.5.1" aria-label="Copy heading and permalink for Exercise 2.2.5.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="solutions"><details id="exercises-Continuous-RVs-1-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-Continuous-RVs-1-2-2-1">
<span class="process-math">\(k = \frac{6}{17}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.2.5.1.a.1"><a href="#exercises-Continuous-RVs-1-2-2" title="Copy heading and permalink for Answer 2.2.5.1.a.1" aria-label="Copy heading and permalink for Answer 2.2.5.1.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 2.2.5.1.a"><a href="#exercises-Continuous-RVs-1-2" title="Copy heading and permalink for Task 2.2.5.1.a" aria-label="Copy heading and permalink for Task 2.2.5.1.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Continuous-RVs-1-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Continuous-RVs-1-3-1-1">Find <span class="process-math">\(\Pr(2 \leq X \leq 3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-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="exercises-Continuous-RVs-1-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-Continuous-RVs-1-3-2-1">
<span class="process-math">\(\Pr(2 \leq X \leq 3) \approx 0.325\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.2.5.1.b.1"><a href="#exercises-Continuous-RVs-1-3-2" title="Copy heading and permalink for Answer 2.2.5.1.b.1" aria-label="Copy heading and permalink for Answer 2.2.5.1.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 2.2.5.1.b"><a href="#exercises-Continuous-RVs-1-3" title="Copy heading and permalink for Task 2.2.5.1.b" aria-label="Copy heading and permalink for Task 2.2.5.1.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 2.2.5.1"><a href="#exercises-Continuous-RVs-1" title="Copy heading and permalink for Exercise 2.2.5.1" aria-label="Copy heading and permalink for Exercise 2.2.5.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Continuous-RVs-2-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find the c.d.f. <span class="process-math">\(F(x)\text{.}\)</span> Use your c.d.f. to find <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Continuous-RVs-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-Continuous-RVs-2-2-1">
<span class="process-math">\(\frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}\text{.}\)</span> <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.2.5.2.1"><a href="#exercises-Continuous-RVs-2-2" title="Copy heading and permalink for Answer 2.2.5.2.1" aria-label="Copy heading and permalink for Answer 2.2.5.2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.2.5.2"><a href="#exercises-Continuous-RVs-2" title="Copy heading and permalink for Exercise 2.2.5.2" aria-label="Copy heading and permalink for Exercise 2.2.5.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Continuous-RVs-3-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([2, 3]\)</span> has c.d.f. <span class="process-math">\(F(x) = \frac{x^3}{3} - x^2 + 4\text{.}\)</span> Find the p.d.f. <span class="process-math">\(f(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-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="exercises-Continuous-RVs-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-Continuous-RVs-3-2-1">
<span class="process-math">\(f(x) = x^2 - 2x.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.2.5.3.1"><a href="#exercises-Continuous-RVs-3-2" title="Copy heading and permalink for Answer 2.2.5.3.1" aria-label="Copy heading and permalink for Answer 2.2.5.3.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.2.5.3"><a href="#exercises-Continuous-RVs-3" title="Copy heading and permalink for Exercise 2.2.5.3" aria-label="Copy heading and permalink for Exercise 2.2.5.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Continuous-RVs-4-1-1">A radioactive material emits particles at an unknown probabilistic rate <span class="process-math">\(\lambda\)</span> particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function <span class="process-math">\(\mathcal{L}(\lambda)\)</span> based on this data. What is the maximum likelihood estimation for <span class="process-math">\(\lambda\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="exercises-Continuous-RVs-4-1-1">Let <span class="process-math">\(T \sim \Exp(3)\text{.}\)</span> Find <span class="process-math">\(\Pr(1 \leq T \leq 3)\)</span> and <span class="process-math">\(\Pr(T \geq 0.5)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 2.2.5.4"><a href="#exercises-Continuous-RVs-4" title="Copy heading and permalink for Exercise 2.2.5.4" aria-label="Copy heading and permalink for Exercise 2.2.5.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Continuous-RVs-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Continuous-RVs-5-1-1">Suppose a parameter <span class="process-math">\(\theta\)</span> takes values in <span class="process-math">\([0, 1]\)</span> with likelihood function <span class="process-math">\(\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-Continuous-RVs-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-Continuous-RVs-4-2-1">
<span class="process-math">\(\Pr(1 \leq T \leq 3) \approx 0.050\text{.}\)</span> <span class="process-math">\(\Pr(T \geq 0.5) \approx 0.223\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Continuous-RVs-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="Exercise 2.2.5.5"><a href="#exercises-Continuous-RVs-5" title="Copy heading and permalink for Exercise 2.2.5.5" aria-label="Copy heading and permalink for Exercise 2.2.5.5">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.2.5"><a href="#exercises-Continuous-RVs" title="Copy heading and permalink for Exercises 2.2.5" aria-label="Copy heading and permalink for Exercises 2.2.5">🔗</a></div></section><div class="autopermalink" data-description="Section 2.2: Continuous Random Variables"><a href="#sec-Continuous-RVs" title="Copy heading and permalink for Section 2.2: Continuous Random Variables" aria-label="Copy heading and permalink for Section 2.2: Continuous Random Variables">🔗</a></div></section></div>
<div class="autopermalink" data-description="Answer 2.2.5.4.1"><a href="#exercises-Continuous-RVs-4-2" title="Copy heading and permalink for Answer 2.2.5.4.1" aria-label="Copy heading and permalink for Answer 2.2.5.4.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.2.5.4"><a href="#exercises-Continuous-RVs-4" title="Copy heading and permalink for Exercise 2.2.5.4" aria-label="Copy heading and permalink for Exercise 2.2.5.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.2.5"><a href="#exercises-Continuous-RVs" title="Copy heading and permalink for Exercises 2.2.5" aria-label="Copy heading and permalink for Exercises 2.2.5">🔗</a></div></section><div class="autopermalink" data-description="Section 2.2: Continuous Random Variables"><a href="#sec-Continuous-RVs" title="Copy heading and permalink for Section 2.2: Continuous Random Variables" aria-label="Copy heading and permalink for Section 2.2: Continuous Random Variables">🔗</a></div></section></div>
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