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knowl/xref/def-binomial-distribution.html
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<article class="definition definition-like"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para">Suppose an event occurs with probability <span class="process-math">\(p\text{.}\)</span> If we perform <span class="process-math">\(n\)</span> independent trials, let <span class="process-math">\(S\)</span> be the random variable which counts the number of trials in which the event occurred. Then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> We’ll write <span class="process-math">\(S \sim \Bin(n, p)\)</span> to denote this. For each value <span class="process-math">\(0 \leq k \leq n\text{,}\)</span> we’ll write <span class="process-math">\(b(k)\)</span> for <span class="process-math">\(\Pr(S = k)\text{.}\)</span> (If we want to keep track of the parameter values, we may write <span class="process-math">\(b(k; n, p)\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#def-binomial-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 2.1.6"><a href="#def-binomial-distribution" title="Copy heading and permalink for Definition 2.1.6" aria-label="Copy heading and permalink for Definition 2.1.6">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#def-binomial-distribution">in-context</a></span>
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knowl/xref/def-continuous-uniform-distribution.html
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knowl/xref/def-discrete-EV.html
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> (also called <dfn class="terminology">mean</dfn>, or <dfn class="terminology">expectation</dfn>) of <span class="process-math">\(X\)</span> is the weighted average of the values of <span class="process-math">\(X\text{,}\)</span> where the weights are the probabilities of <span class="process-math">\(X\)</span> taking each value:</div>
<div class="displaymath process-math">
\begin{gather*}
\E(X) = \sum_{i=1}^n x_i \Pr(X = x_i).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 3.1.1"><a href="#def-discrete-EV" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#def-discrete-EV">in-context</a></span>
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knowl/xref/def-variance.html
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. The <dfn class="terminology">variance</dfn> of <span class="process-math">\(X\)</span> is:</div>
<div class="displaymath process-math">
\begin{gather*}
\Var(X) = \E\left[ \left(X - \E(X)\right)^2 \right].
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 3.2.1"><a href="#def-variance" title="Copy heading and permalink for Definition 3.2.1" aria-label="Copy heading and permalink for Definition 3.2.1">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#def-variance">in-context</a></span>
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knowl/xref/example-EV-die.html
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<article class="example example-like"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
<div class="displaymath process-math">
\begin{align*}
\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
\end{align*}
</div>
<div class="para">For example, the expected value of a fair 6-sided die roll <span class="process-math">\(R\)</span> is:</div>
<div class="displaymath process-math">
\begin{align*}
\E(R) \amp = 1 \left(\frac{1}{6}\right) + 2 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{6}\right) + 4 \left(\frac{1}{6}\right) + 5 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{6}\right) \\
\amp = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-die">in-context</a></span>
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knowl/xref/example-EV-sum-2-dice.html
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k \cdot \Pr(S = k)\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(20/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(42/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(40/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(36/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">10</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">11</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(22/36\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">12</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
</tr>
</table></div></figure> <div class="para">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-sum-2-dice">in-context</a></span>
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knowl/xref/example-binomial-EV.html
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<article class="example example-like"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">Suppose <span class="process-math">\(S \sim \Bin(n, p)\text{.}\)</span> <a href="sec-Discrete-RVs.html#def-binomial-distribution" class="xref" data-knowl="./knowl/xref/def-binomial-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.1.6">Definition 2.1.6</a> describes a binomially distributed random variable as counting the number of occurrences of some event, which either happens or not in each of <span class="process-math">\(n\)</span> independent trials. This leads us to a very natural idea: identify each individual possible occurrence of the event, and assign it an indicator random variable. In this case, let <span class="process-math">\(H_i\)</span> indicate that flip <span class="process-math">\(i\)</span> comes up heads. Then</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html">
\begin{gather*}
S = H_1 + H_2 + \dotsb + H_n.
\end{gather*}
</div>
<div class="para">We already know (<a href="sec-Expected-Value.html#example-indicator-EV" class="xref" data-knowl="./knowl/xref/example-indicator-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.5">Example 3.1.5</a>) that the expected value of each <span class="process-math">\(H_i\)</span> is the probability of the indicated event—that flip <span class="process-math">\(i\)</span> is heads. This is precisely the parameter <span class="process-math">\(p\text{.}\)</span> Therefore:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html">
\begin{align*}
\E(S) \amp = \E(H_1 + H_2 + \dotsb + H_n) \\
\amp = \E(H_1) + \E(H_2) + \dotsb + \E(H_n) \\
\amp = \underbrace{p + p + \dotsb + p}_{n \text{ times}} \\
\amp = np.
\end{align*}
</div>
<div class="para">Remember this formula! We’ll make frequent use of it.</div>
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<div class="autopermalink" data-description="Example 3.1.11"><a href="#example-binomial-EV" title="Copy heading and permalink for Example 3.1.11" aria-label="Copy heading and permalink for Example 3.1.11">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-binomial-EV">in-context</a></span>
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knowl/xref/example-indicator-EV.html
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> Then:</div>
<div class="displaymath process-math">
\begin{gather*}
\E(X) = 0 \cdot (1-p) + 1 \cdot p = p.
\end{gather*}
</div>
<div class="para">That is, the expected value of an indicator random variable is the probability of the event that it indicates.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 3.1.5"><a href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-indicator-EV">in-context</a></span>
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knowl/xref/example-indicator-variance.html
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> In <a href="sec-Expected-Value.html#example-indicator-EV" class="xref" data-knowl="./knowl/xref/example-indicator-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.5">Example 3.1.5</a>, we saw <span class="process-math">\(\E(X) = p\text{.}\)</span> Notice that <span class="process-math">\(X\)</span> takes the values 0 and 1 (with probabilities <span class="process-math">\(1 - p\)</span> and <span class="process-math">\(p\text{,}\)</span> respectively). But <span class="process-math">\(0^2 = 0\text{,}\)</span> and <span class="process-math">\(1^1 = 1\text{,}\)</span> so the expected value calculation for <span class="process-math">\(X^2\)</span> is precisely the same as for <span class="process-math">\(X\text{!}\)</span>
</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html">
\begin{align*}
\E\left(X^2\right) \amp = 0^2 \cdot (1-p) + 1^2 \cdot p = p.
\end{align*}
</div>
<div class="para">So:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html">
\begin{align*}
\Var(X) \amp \E\left(X^2\right) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p).
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 3.2.6"><a href="#example-indicator-variance" title="Copy heading and permalink for Example 3.2.6" aria-label="Copy heading and permalink for Example 3.2.6">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#example-indicator-variance">in-context</a></span>
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knowl/xref/example-joint-indicators.html
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<article class="example example-like"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para">A poll asks two yes/no questions. Let <span class="process-math">\(X\)</span> indicate a yes on Question 1 and <span class="process-math">\(Y\)</span> indicate a yes on Question 2. When the data is collected, the following joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is created:<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution for Indicator Random Variables<div class="autopermalink" data-description="Table 2.3.3: Joint Distribution for Indicator Random Variables"><a href="#example-joint-indicators-1-2" title="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables" aria-label="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.1</td>
<td class="c m b1 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.3</td>
<td class="c m b0 r0 l0 t0 lines">0.4</td>
</tr>
</table></div></figure> <div class="para logical">
<div class="para">Note that the sum of all values in the table is 1; this is a probability distribution, and total probability must be 1. This table doesn’t show the distributions for <span class="process-math">\(X\)</span> or <span class="process-math">\(Y\)</span> individually. The data for the two distributions is mixed together. We can, if we wish, take this information and determine separate distributions for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span> For the distribution for <span class="process-math">\(X\text{,}\)</span> we need to know <span class="process-math">\(\Pr(X = 0)\)</span> and <span class="process-math">\(\Pr(X = 1)\)</span> (with no reference to <span class="process-math">\(Y\)</span>). We can get these probabilities by summing along the columns of the table:</div>
<div class="displaymath process-math">
\begin{align*}
\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\
\Pr(X = 1) \amp = 0.2 + 0.4 = 0.6
\end{align*}
</div>
<div class="para">To get the distribution for <span class="process-math">\(Y\text{,}\)</span> we should add along the rows:</div>
<div class="displaymath process-math">
\begin{align*}
\Pr(Y = 0) \amp = 0.1 + 0.2 = 0.3 \\
\Pr(Y = 1) \amp = 0.3 + 0.4 = 0.7
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 2.3.2"><a href="#example-joint-indicators" title="Copy heading and permalink for Example 2.3.2" aria-label="Copy heading and permalink for Example 2.3.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Joint-Distributions.html#example-joint-indicators">in-context</a></span>
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knowl/xref/example-poisson-rate.html
Regular → Executable
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knowl/xref/example-rolls-conditional.html
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knowl/xref/example-sample-space.html
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<article class="example example-like"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para">What would the sample space look like if we roll the die two times and recorded the results?<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para logical">
<div class="displaymath process-math">
\begin{align*}
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
\end{align*}
</div>
<div class="para">Note that, for example, <span class="process-math">\((1, 2)\)</span> is a different outcome from <span class="process-math">\((2, 1)\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 1.1.2.1"><a href="#example-sample-space-2" title="Copy heading and permalink for Answer 1.1.2.1" aria-label="Copy heading and permalink for Answer 1.1.2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Example 1.1.2"><a href="#example-sample-space" title="Copy heading and permalink for Example 1.1.2" aria-label="Copy heading and permalink for Example 1.1.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Set-Theory.html#example-sample-space">in-context</a></span>
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knowl/xref/example-uniform-EV.html
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<article class="example example-like"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> be uniform on the interval <span class="process-math">\([a, b]\text{.}\)</span> So <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span> We might reasonably expect that the average value of <span class="process-math">\(X\)</span> would be the midpoint of the interval. Let’s check that:</div>
<div class="displaymath process-math">
\begin{align*}
\E(X) \amp = \int_a^b x f(x)\ dx \\
\amp = \int_a^b \frac{x}{b - a} \ dx \\
\amp = \frac{1}{b-a} \left(\frac{x^2}{2}\right)\bigg|_a^b \\
\amp = \frac{1}{b-a} \left( \frac{b^2}{2} - \frac{a^2}{2} \right) \\
\amp = \frac{1}{b - a}\left( \frac{(b - a)(b + a)}{2}\right) \\
\amp = \frac{a + b}{2}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#example-uniform-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 3.1.7"><a href="#example-uniform-EV" title="Copy heading and permalink for Example 3.1.7" aria-label="Copy heading and permalink for Example 3.1.7">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-uniform-EV">in-context</a></span>
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knowl/xref/fact-binomial-probability-formula.html
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knowl/xref/table-binomial-example.html
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knowl/xref/thm-variance-formula.html
Executable
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<article class="theorem theorem-like"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical">
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. Then:</div>
<div class="displaymath process-math">
\begin{gather*}
\Var(X) = \E\left(X^2\right) - \left(\E(X)\right)^2.
\end{gather*}
</div>
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