46 lines
3.6 KiB
HTML
Executable File
46 lines
3.6 KiB
HTML
Executable File
<!DOCTYPE html>
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<html lang="en-US" dir="ltr">
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<meta name="robots" content="noindex, nofollow">
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</head>
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<body class="ignore-math">
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<article class="example example-like"><h4 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para logical">
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<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>
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<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html">
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\begin{gather*}
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S = H_1 + H_2 + \dotsb + H_n.
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\end{gather*}
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</div>
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<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>
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<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html">
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\begin{align*}
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\E(S) \amp = \E(H_1 + H_2 + \dotsb + H_n) \\
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\amp = \E(H_1) + \E(H_2) + \dotsb + \E(H_n) \\
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\amp = \underbrace{p + p + \dotsb + p}_{n \text{ times}} \\
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\amp = np.
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\end{align*}
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</div>
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<div class="para">Remember this formula! We’ll make frequent use of it.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#example-binomial-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>
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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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</body>
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</html>
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