File: The-Chi_002dsquared-Distribution.html

package info (click to toggle)
gsl-ref-html 2.3-1
links: PTS
area: non-free
in suites: bookworm, bullseye, buster, forky, sid, trixie
size: 6,876 kB
ctags: 4,574
sloc: makefile: 35
file content (120 lines) | stat: -rw-r--r-- 5,928 bytes
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".

(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library &ndash; Reference Manual: The Chi-squared Distribution</title>

<meta name="description" content="GNU Scientific Library &ndash; Reference Manual: The Chi-squared Distribution">
<meta name="keywords" content="GNU Scientific Library &ndash; Reference Manual: The Chi-squared Distribution">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Random-Number-Distributions.html#Random-Number-Distributions" rel="up" title="Random Number Distributions">
<link href="The-F_002ddistribution.html#The-F_002ddistribution" rel="next" title="The F-distribution">
<link href="The-Lognormal-Distribution.html#The-Lognormal-Distribution" rel="previous" title="The Lognormal Distribution">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="The-Chi_002dsquared-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-F_002ddistribution.html#The-F_002ddistribution" accesskey="n" rel="next">The F-distribution</a>, Previous: <a href="The-Lognormal-Distribution.html#The-Lognormal-Distribution" accesskey="p" rel="previous">The Lognormal Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-Chi_002dsquared-Distribution-1"></a>
<h3 class="section">20.18 The Chi-squared Distribution</h3>
<p>The chi-squared distribution arises in statistics.  If <em>Y_i</em> are
<em>n</em> independent Gaussian random variates with unit variance then the
sum-of-squares,
</p>
<div class="example">
<pre class="example">X_i = \sum_i Y_i^2
</pre></div>

<p>has a chi-squared distribution with <em>n</em> degrees of freedom.
</p>
<dl>
<dt><a name="index-gsl_005fran_005fchisq"></a>Function: <em>double</em> <strong>gsl_ran_chisq</strong> <em>(const gsl_rng * <var>r</var>, double <var>nu</var>)</em></dt>
<dd><a name="index-Chi_002dsquared-distribution"></a>
<p>This function returns a random variate from the chi-squared distribution
with <var>nu</var> degrees of freedom. The distribution function is,
</p>
<div class="example">
<pre class="example">p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
</pre></div>

<p>for <em>x &gt;= 0</em>. 
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fchisq_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_chisq_pdf</strong> <em>(double <var>x</var>, double <var>nu</var>)</em></dt>
<dd><p>This function computes the probability density <em>p(x)</em> at <var>x</var>
for a chi-squared distribution with <var>nu</var> degrees of freedom, using
the formula given above.
</p></dd></dl>

<br>

<dl>
<dt><a name="index-gsl_005fcdf_005fchisq_005fP"></a>Function: <em>double</em> <strong>gsl_cdf_chisq_P</strong> <em>(double <var>x</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fchisq_005fQ"></a>Function: <em>double</em> <strong>gsl_cdf_chisq_Q</strong> <em>(double <var>x</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fchisq_005fPinv"></a>Function: <em>double</em> <strong>gsl_cdf_chisq_Pinv</strong> <em>(double <var>P</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fchisq_005fQinv"></a>Function: <em>double</em> <strong>gsl_cdf_chisq_Qinv</strong> <em>(double <var>Q</var>, double <var>nu</var>)</em></dt>
<dd><p>These functions compute the cumulative distribution functions
<em>P(x)</em>, <em>Q(x)</em> and their inverses for the chi-squared
distribution with <var>nu</var> degrees of freedom.
</p></dd></dl>






</body>
</html>