41 lines
3.3 KiB
HTML
41 lines
3.3 KiB
HTML
<!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"><h2 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">3.2.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h2>
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<div class="para logical">
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<div class="para">Recall the setup from <a href="sec-Expected-Value.html#example-binomial-EV" class="xref" data-knowl="./knowl/xref/example-binomial-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.11">Example 3.1.11</a>: a binomial random variable <span class="process-math">\(S\)</span> is the sum <span class="process-math">\(H_1 + H_2 + \dotsb + H_n\)</span> of <span class="process-math">\(n\)</span> indicator random variables, each with parameter <span class="process-math">\(p\text{.}\)</span> The results of different coin flips are independent from each other, so the random variables <span class="process-math">\(H_1, \dotsc, H_n\)</span> are independent from each other. From <a href="sec-Variance.html#example-indicator-variance" class="xref" data-knowl="./knowl/xref/example-indicator-variance.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.2.6">Example 3.2.6</a>, we know that <span class="process-math">\(\Var(H_i) = p(1-p)\)</span> for each <span class="process-math">\(i\text{.}\)</span> Therefore:</div>
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<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-EV.html ./knowl/xref/example-indicator-variance.html">
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\begin{align*}
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\Var(S) \amp = \Var(H_1 + H_2 + \dotsb + H_n) \\
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\amp = \Var(H_1) + \Var(H_2) + \dotsb + \Var(H_n) \\
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\amp = \underbrace{p(1-p) + p(1-p) + \dotsb + p(1-p)}_{n \text{ times}} \\
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\amp = np(1-p).
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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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-binomial-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>
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<div class="autopermalink" aria-hidden="true" data-description="Example 3.2.7"><a tabindex="-1" href="#example-binomial-variance" title="Copy heading and permalink for Example 3.2.7" aria-label="Copy heading and permalink for Example 3.2.7">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Variance.html#example-binomial-variance">In Context</a></span>
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</body>
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</html>
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