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

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<a name="The-t_002ddistribution"></a>
<div class="header">
<p>
Next: <a href="The-Beta-Distribution.html#The-Beta-Distribution" accesskey="n" rel="next">The Beta Distribution</a>, Previous: <a href="The-F_002ddistribution.html#The-F_002ddistribution" accesskey="p" rel="previous">The F-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-t_002ddistribution-1"></a>
<h3 class="section">20.20 The t-distribution</h3>
<p>The t-distribution arises in statistics.  If <em>Y_1</em> has a normal
distribution and <em>Y_2</em> has a chi-squared distribution with
<em>\nu</em> degrees of freedom then the ratio,
</p>
<div class="example">
<pre class="example">X = { Y_1 \over \sqrt{Y_2 / \nu} }
</pre></div>

<p>has a t-distribution <em>t(x;\nu)</em> with <em>\nu</em> degrees of freedom.
</p>
<dl>
<dt><a name="index-gsl_005fran_005ftdist"></a>Function: <em>double</em> <strong>gsl_ran_tdist</strong> <em>(const gsl_rng * <var>r</var>, double <var>nu</var>)</em></dt>
<dd><a name="index-t_002ddistribution"></a>
<a name="index-Student-t_002ddistribution"></a>
<p>This function returns a random variate from the t-distribution.  The
distribution function is,
</p>
<div class="example">
<pre class="example">p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
   (1 + x^2/\nu)^{-(\nu + 1)/2} dx
</pre></div>

<p>for <em>-\infty &lt; x &lt; +\infty</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005ftdist_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_tdist_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 t-distribution with <var>nu</var> degrees of freedom, using the formula
given above.
</p></dd></dl>

<br>

<dl>
<dt><a name="index-gsl_005fcdf_005ftdist_005fP"></a>Function: <em>double</em> <strong>gsl_cdf_tdist_P</strong> <em>(double <var>x</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ftdist_005fQ"></a>Function: <em>double</em> <strong>gsl_cdf_tdist_Q</strong> <em>(double <var>x</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ftdist_005fPinv"></a>Function: <em>double</em> <strong>gsl_cdf_tdist_Pinv</strong> <em>(double <var>P</var>, double <var>nu</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005ftdist_005fQinv"></a>Function: <em>double</em> <strong>gsl_cdf_tdist_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 t-distribution
with <var>nu</var> degrees of freedom.
</p></dd></dl>




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