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<title>The t-distribution - GNU Scientific Library -- Reference Manual</title>
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<a name="The-t-distribution"></a>
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Next: <a rel="next" accesskey="n" href="The-Beta-Distribution.html">The Beta Distribution</a>,
Previous: <a rel="previous" accesskey="p" href="The-F_002ddistribution.html">The F-distribution</a>,
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<h3 class="section">20.19 The t-distribution</h3>
<p>The t-distribution arises in statistics. If Y_1 has a normal
distribution and Y_2 has a chi-squared distribution with
\nu degrees of freedom then the ratio,
<pre class="example"> X = { Y_1 \over \sqrt{Y_2 / \nu} }
</pre>
<p class="noindent">has a t-distribution t(x;\nu) with \nu degrees of freedom.
<div class="defun">
— Function: double <b>gsl_ran_tdist</b> (<var>const gsl_rng * r, double nu</var>)<var><a name="index-gsl_005fran_005ftdist-1788"></a></var><br>
<blockquote><p><a name="index-t_002ddistribution-1789"></a><a name="index-Student-t_002ddistribution-1790"></a>This function returns a random variate from the t-distribution. The
distribution function is,
<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>
<p class="noindent">for -\infty < x < +\infty.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_ran_tdist_pdf</b> (<var>double x, double nu</var>)<var><a name="index-gsl_005fran_005ftdist_005fpdf-1791"></a></var><br>
<blockquote><p>This function computes the probability density p(x) at <var>x</var>
for a t-distribution with <var>nu</var> degrees of freedom, using the formula
given above.
</p></blockquote></div>
<pre class="sp">
</pre>
<div class="defun">
— Function: double <b>gsl_cdf_tdist_P</b> (<var>double x, double nu</var>)<var><a name="index-gsl_005fcdf_005ftdist_005fP-1792"></a></var><br>
— Function: double <b>gsl_cdf_tdist_Q</b> (<var>double x, double nu</var>)<var><a name="index-gsl_005fcdf_005ftdist_005fQ-1793"></a></var><br>
— Function: double <b>gsl_cdf_tdist_Pinv</b> (<var>double P, double nu</var>)<var><a name="index-gsl_005fcdf_005ftdist_005fPinv-1794"></a></var><br>
— Function: double <b>gsl_cdf_tdist_Qinv</b> (<var>double Q, double nu</var>)<var><a name="index-gsl_005fcdf_005ftdist_005fQinv-1795"></a></var><br>
<blockquote><p>These functions compute the cumulative distribution functions
P(x), Q(x) and their inverses for the t-distribution
with <var>nu</var> degrees of freedom.
</p></blockquote></div>
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