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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">
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<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>
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<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>
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<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>
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</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>
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<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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