andy/Probability-and-Statistics-Notes
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Latest build deployed.

Browse Source
This commit is contained in:
andy
2026-01-11 13:24:27 -05:00
parent 5f05bd0b9f
commit 63ce1d952b
674 changed files with 4797 additions and 562 deletions
Download Patch File Download Diff File Expand all files Collapse all files
knowl/xref/example-EV-sum-2-dice.html
Executable
+90
View File
@@ -0,0 +1,90 @@
<!DOCTYPE html>
<html lang="en-US" dir="ltr">
<!--******************************************-->
<!--* Authored with PreTeXt *-->
<!--* pretextbook.org *-->
<!--* Theme: boulder *-->
<!--* Palette: *-->
<!--******************************************-->
<!--********************************************-->
<!--* Generated from PreTeXt source *-->
<!--* *-->
<!--* https://pretextbook.org *-->
<!--* *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<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">3.1.3</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>
</div>
<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>
</body>
</html>