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<a name="The-Dirichlet-Distribution"></a>
<div class="header">
<p>
Next: <a href="General-Discrete-Distributions.html#General-Discrete-Distributions" accesskey="n" rel="next">General Discrete Distributions</a>, Previous: <a href="The-Type_002d2-Gumbel-Distribution.html#The-Type_002d2-Gumbel-Distribution" accesskey="p" rel="previous">The Type-2 Gumbel 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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<a name="The-Dirichlet-Distribution-1"></a>
<h3 class="section">20.28 The Dirichlet Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005fdirichlet"></a>Function: <em>void</em> <strong>gsl_ran_dirichlet</strong> <em>(const gsl_rng * <var>r</var>, size_t <var>K</var>, const double <var>alpha</var>[], double <var>theta</var>[])</em></dt>
<dd><a name="index-Dirichlet-distribution"></a>
<p>This function returns an array of <var>K</var> random variates from a Dirichlet
distribution of order <var>K</var>-1. The distribution function is
</p>
<div class="example">
<pre class="example">p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K = 
  (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K
</pre></div>

<p>for <em>theta_i &gt;= 0</em>
and <em>alpha_i &gt; 0</em>.  The delta function ensures that <em>\sum \theta_i = 1</em>.
The normalization factor <em>Z</em> is
</p>
<div class="example">
<pre class="example">Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}
</pre></div>

<p>The random variates are generated by sampling <var>K</var> values 
from gamma distributions with parameters 
<em>a=alpha_i, b=1</em>, 
and renormalizing. 
See A.M. Law, W.D. Kelton, <cite>Simulation Modeling and Analysis</cite> (1991).
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fdirichlet_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_dirichlet_pdf</strong> <em>(size_t <var>K</var>, const double <var>alpha</var>[], const double <var>theta</var>[]) </em></dt>
<dd><p>This function computes the probability density 
<em>p(\theta_1, ... , \theta_K)</em>
at <var>theta</var>[<var>K</var>] for a Dirichlet distribution with parameters 
<var>alpha</var>[<var>K</var>], using the formula given above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fdirichlet_005flnpdf"></a>Function: <em>double</em> <strong>gsl_ran_dirichlet_lnpdf</strong> <em>(size_t <var>K</var>, const double <var>alpha</var>[], const double <var>theta</var>[]) </em></dt>
<dd><p>This function computes the logarithm of the probability density 
<em>p(\theta_1, ... , \theta_K)</em>
for a Dirichlet distribution with parameters 
<var>alpha</var>[<var>K</var>].
</p></dd></dl>




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