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@@ -511,27 +511,67 @@ p(k; \lambda, t) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}.
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.5</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-Discrete-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-Discrete-RVs-1-1-1">We flip a coin with unknown bias 100 times and observe 43 heads. What is the maximum likelihood estimation for the probability of the coin coming up heads?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="exercises-Discrete-RVs-1-1-1">An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability <span class="process-math">\(p = 0.3\text{,}\)</span> find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-Discrete-RVs-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="exercises-Discrete-RVs-1-2-1">
<div class="para">Let <span class="process-math">\(S\)</span> be the number of heads. Then <span class="process-math">\(S \sim \Bin(20, 0.3)\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="exercises-Discrete-RVs-1-2-1-3">
\begin{align*}
\Pr(S = 5) \amp = {20 \choose 5} (0.3)^5 (0.7)^{20 - 5} \\
\amp = \frac{20!}{(5!)(15!)} (0.3)^5 (0.7)^{15} \\
\amp = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} (0.3)^5 (0.7)^{15} \\
\amp = (19 \times 3 \times 17 \times 16) (0.3)^5 (0.7)^{15} \\
\amp \approx 0.179
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-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 2.1.5.1.1"><a href="#exercises-Discrete-RVs-1-2" title="Copy heading and permalink for Solution 2.1.5.1.1" aria-label="Copy heading and permalink for Solution 2.1.5.1.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.1.5.1"><a href="#exercises-Discrete-RVs-1" title="Copy heading and permalink for Exercise 2.1.5.1" aria-label="Copy heading and permalink for Exercise 2.1.5.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-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-Discrete-RVs-2-1-1">An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability <span class="process-math">\(p = 0.3\text{,}\)</span> find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="introduction" id="exercises-Discrete-RVs-2-1"><div class="para" id="exercises-Discrete-RVs-2-1-1">An experiment consists of flipping a coin repeatedly until we first see heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-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="exercises-Discrete-RVs-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Discrete-RVs-2-2-1-1">If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 2.1.5.2"><a href="#exercises-Discrete-RVs-2" title="Copy heading and permalink for Exercise 2.1.5.2" aria-label="Copy heading and permalink for Exercise 2.1.5.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-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-Discrete-RVs-3-1-1">An experiment consists of flipping a coin repeatedly until we first see heads.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-Discrete-RVs-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 logical" id="exercises-Discrete-RVs-2-2-2-1">
<div class="para">Let <span class="process-math">\(T\)</span> be the number of flips until we see heads. Then <span class="process-math">\(T\)</span> is geometric with parameter <span class="process-math">\(p = 0.4\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="exercises-Discrete-RVs-2-2-2-1-4">
\begin{align*}
\Pr(T = k) \amp = (1-0.4)^{k-1}(0.4) = 0.6^{k-1} \cdot 0.4
\end{align*}
</div>
<div class="autopermalink" data-description="Exercise 2.1.5.3"><a href="#exercises-Discrete-RVs-3" title="Copy heading and permalink for Exercise 2.1.5.3" aria-label="Copy heading and permalink for Exercise 2.1.5.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-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-Discrete-RVs-4-1-1">If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-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="Exercise 2.1.5.4"><a href="#exercises-Discrete-RVs-4" title="Copy heading and permalink for Exercise 2.1.5.4" aria-label="Copy heading and permalink for Exercise 2.1.5.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-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-Discrete-RVs-5-1-1">Suppose the coin has an unknown probability of coming up heads. We perform the experiment in five independent trials, during which it takes 4, 5, 4, 3, and 6 flips to see our first heads in each trial. What is the maximum likelihood estimation for the probability of the coin coming up heads on a flip?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Solution 2.1.5.2.a.1"><a href="#exercises-Discrete-RVs-2-2-2" title="Copy heading and permalink for Solution 2.1.5.2.a.1" aria-label="Copy heading and permalink for Solution 2.1.5.2.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 2.1.5.2.a"><a href="#exercises-Discrete-RVs-2-2" title="Copy heading and permalink for Task 2.1.5.2.a" aria-label="Copy heading and permalink for Task 2.1.5.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Discrete-RVs-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Discrete-RVs-2-3-1-1">Which flip has the highest chance of being the first flip to come up heads?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 2.1.5.5"><a href="#exercises-Discrete-RVs-5" title="Copy heading and permalink for Exercise 2.1.5.5" aria-label="Copy heading and permalink for Exercise 2.1.5.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Discrete-RVs-6-1-1">A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-Discrete-RVs-2-3-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para" id="exercises-Discrete-RVs-2-3-2-1">
<span class="process-math">\(\Pr(T = k) = 0.6^{k-1} \cdot 0.4\text{,}\)</span> so every additional flip multiplies the probability by 0.6. Therefore, the highest value for <span class="process-math">\(\Pr(T = k)\)</span> occurs when <span class="process-math">\(k = 1\text{,}\)</span> in which case <span class="process-math">\(\Pr(T = 1) = 0.4.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-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="Exercise 2.1.5.6"><a href="#exercises-Discrete-RVs-6" title="Copy heading and permalink for Exercise 2.1.5.6" aria-label="Copy heading and permalink for Exercise 2.1.5.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-RVs-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Discrete-RVs-7-1-1">A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" data-description="Solution 2.1.5.2.b.1"><a href="#exercises-Discrete-RVs-2-3-2" title="Copy heading and permalink for Solution 2.1.5.2.b.1" aria-label="Copy heading and permalink for Solution 2.1.5.2.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 2.1.5.2.b"><a href="#exercises-Discrete-RVs-2-3" title="Copy heading and permalink for Task 2.1.5.2.b" aria-label="Copy heading and permalink for Task 2.1.5.2.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 2.1.5.2"><a href="#exercises-Discrete-RVs-2" title="Copy heading and permalink for Exercise 2.1.5.2" aria-label="Copy heading and permalink for Exercise 2.1.5.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Discrete-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-Discrete-RVs-3-1-1">A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-RVs-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 2.1.5.7"><a href="#exercises-Discrete-RVs-7" title="Copy heading and permalink for Exercise 2.1.5.7" aria-label="Copy heading and permalink for Exercise 2.1.5.7">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.1.5"><a href="#exercises-Discrete-RVs" title="Copy heading and permalink for Exercises 2.1.5" aria-label="Copy heading and permalink for Exercises 2.1.5">🔗</a></div></section><div class="autopermalink" data-description="Section 2.1: Discrete Random Variables"><a href="#sec-Discrete-RVs" title="Copy heading and permalink for Section 2.1: Discrete Random Variables" aria-label="Copy heading and permalink for Section 2.1: Discrete Random Variables">🔗</a></div></section></div>
<div class="solutions"><details id="exercises-Discrete-RVs-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="exercises-Discrete-RVs-3-2-1">
<div class="para">Let <span class="process-math">\(N\)</span> be the number of customers seen during the afternoon shift. Since the store averages 20 customers per hour, it will average 80 per 4-hours. So <span class="process-math">\(N \sim \Poiss(20, 4)\)</span> will have distribution <span class="process-math">\(\Pr(N = k) = \frac{80^{k}}{k!} e^{-80}\text{.}\)</span> Therefore:</div>
<div class="displaymath process-math" id="exercises-Discrete-RVs-3-2-1-4">
\begin{gather*}
\Pr(N = 80) = \frac{80^{80}}{80!} e^{-80} \approx 0.045.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Discrete-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="Solution 2.1.5.3.1"><a href="#exercises-Discrete-RVs-3-2" title="Copy heading and permalink for Solution 2.1.5.3.1" aria-label="Copy heading and permalink for Solution 2.1.5.3.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2.1.5.3"><a href="#exercises-Discrete-RVs-3" title="Copy heading and permalink for Exercise 2.1.5.3" aria-label="Copy heading and permalink for Exercise 2.1.5.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.1.5"><a href="#exercises-Discrete-RVs" title="Copy heading and permalink for Exercises 2.1.5" aria-label="Copy heading and permalink for Exercises 2.1.5">🔗</a></div></section><div class="autopermalink" data-description="Section 2.1: Discrete Random Variables"><a href="#sec-Discrete-RVs" title="Copy heading and permalink for Section 2.1: Discrete Random Variables" aria-label="Copy heading and permalink for Section 2.1: Discrete Random Variables">🔗</a></div></section></div>
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