42 lines
2.5 KiB
HTML
42 lines
2.5 KiB
HTML
<!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"><h2 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h2>
|
||
<div class="para logical">
|
||
<div class="para">Let <span class="process-math">\(X\)</span> be uniform on the interval <span class="process-math">\([a, b]\text{.}\)</span> So <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span> We might reasonably expect that the average value of <span class="process-math">\(X\)</span> would be the midpoint of the interval. Let’s check that:</div>
|
||
<div class="displaymath process-math">
|
||
\begin{align*}
|
||
\E(X) \amp = \int_a^b x f(x)\ dx \\
|
||
\amp = \int_a^b \frac{x}{b - a} \ dx \\
|
||
\amp = \frac{1}{b-a} \left(\frac{x^2}{2}\right)\bigg|_a^b \\
|
||
\amp = \frac{1}{b-a} \left( \frac{b^2}{2} - \frac{a^2}{2} \right) \\
|
||
\amp = \frac{1}{b - a}\left( \frac{(b - a)(b + a)}{2}\right) \\
|
||
\amp = \frac{a + b}{2}.
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-uniform-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.7"><a tabindex="-1" href="#example-uniform-EV" title="Copy heading and permalink for Example 3.1.7" aria-label="Copy heading and permalink for Example 3.1.7">🔗</a></div>
|
||
</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-uniform-EV">In Context</a></span>
|
||
</body>
|
||
</html>
|