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<title>GNU Scientific Library &ndash; Reference Manual: The F-distribution</title>

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<a name="The-F_002ddistribution"></a>
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<p>
Next: <a href="The-t_002ddistribution.html#The-t_002ddistribution" accesskey="n" rel="next">The t-distribution</a>, Previous: <a href="The-Chi_002dsquared-Distribution.html#The-Chi_002dsquared-Distribution" accesskey="p" rel="previous">The Chi-squared 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>
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<hr>
<a name="The-F_002ddistribution-1"></a>
<h3 class="section">20.19 The F-distribution</h3>
<p>The F-distribution arises in statistics.  If <em>Y_1</em> and <em>Y_2</em>
are chi-squared deviates with <em>\nu_1</em> and <em>\nu_2</em> degrees of
freedom then the ratio,
</p>
<div class="example">
<pre class="example">X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
</pre></div>

<p>has an F-distribution <em>F(x;\nu_1,\nu_2)</em>.
</p>
<dl>
<dt><a name="index-gsl_005fran_005ffdist"></a>Function: <em>double</em> <strong>gsl_ran_fdist</strong> <em>(const gsl_rng * <var>r</var>, double <var>nu1</var>, double <var>nu2</var>)</em></dt>
<dd><a name="index-F_002ddistribution"></a>
<p>This function returns a random variate from the F-distribution with degrees of freedom <var>nu1</var> and <var>nu2</var>. The distribution function is,
</p>
<div class="example">
<pre class="example">p(x) dx = 
   { \Gamma((\nu_1 + \nu_2)/2)
        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 
   \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 
   x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
</pre></div>

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

<dl>
<dt><a name="index-gsl_005fran_005ffdist_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_fdist_pdf</strong> <em>(double <var>x</var>, double <var>nu1</var>, double <var>nu2</var>)</em></dt>
<dd><p>This function computes the probability density <em>p(x)</em> at <var>x</var>
for an F-distribution with <var>nu1</var> and <var>nu2</var> degrees of freedom,
using the formula given above.
</p></dd></dl>

<br>

<dl>
<dt><a name="index-gsl_005fcdf_005ffdist_005fP"></a>Function: <em>double</em> <strong>gsl_cdf_fdist_P</strong> <em>(double <var>x</var>, double <var>nu1</var>, double <var>nu2</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ffdist_005fQ"></a>Function: <em>double</em> <strong>gsl_cdf_fdist_Q</strong> <em>(double <var>x</var>, double <var>nu1</var>, double <var>nu2</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ffdist_005fPinv"></a>Function: <em>double</em> <strong>gsl_cdf_fdist_Pinv</strong> <em>(double <var>P</var>, double <var>nu1</var>, double <var>nu2</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ffdist_005fQinv"></a>Function: <em>double</em> <strong>gsl_cdf_fdist_Qinv</strong> <em>(double <var>Q</var>, double <var>nu1</var>, double <var>nu2</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 F-distribution
with <var>nu1</var> and <var>nu2</var> degrees of freedom.
</p></dd></dl>




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