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<title>GNU Scientific Library &ndash; Reference Manual: The Chi-squared Distribution</title>

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<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>






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