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<body class="ignore-math">
<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">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>
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<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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