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<a name="The-Multinomial-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-Negative-Binomial-Distribution.html#The-Negative-Binomial-Distribution" accesskey="n" rel="next">The Negative Binomial Distribution</a>, Previous: <a href="The-Binomial-Distribution.html#The-Binomial-Distribution" accesskey="p" rel="previous">The Binomial 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-Multinomial-Distribution-1"></a>
<h3 class="section">20.33 The Multinomial Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005fmultinomial"></a>Function: <em>void</em> <strong>gsl_ran_multinomial</strong> <em>(const gsl_rng * <var>r</var>, size_t <var>K</var>, unsigned int <var>N</var>, const double <var>p</var>[], unsigned int <var>n</var>[])</em></dt>
<dd><a name="index-Multinomial-distribution"></a>

<p>This function computes a random sample <var>n</var>[] from the multinomial
distribution formed by <var>N</var> trials from an underlying distribution
<var>p</var>[<var>K</var>]. The distribution function for <var>n</var>[] is,
</p>
<div class="example">
<pre class="example">P(n_1, n_2, ..., n_K) = 
  (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K
</pre></div>

<p>where <em>(n_1, n_2, ..., n_K)</em> 
are nonnegative integers with 
<em>sum_{k=1}^K n_k = N</em>,
and
<em>(p_1, p_2, ..., p_K)</em>
is a probability distribution with <em>\sum p_i = 1</em>.  
If the array <var>p</var>[<var>K</var>] is not normalized then its entries will be
treated as weights and normalized appropriately.  The arrays <var>n</var>[]
and <var>p</var>[] must both be of length <var>K</var>.
</p>
<p>Random variates are generated using the conditional binomial method (see
C.S. Davis, <cite>The computer generation of multinomial random
variates</cite>, Comp. Stat. Data Anal. 16 (1993) 205&ndash;217 for details).
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fmultinomial_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_multinomial_pdf</strong> <em>(size_t <var>K</var>, const double <var>p</var>[], const unsigned int <var>n</var>[]) </em></dt>
<dd><p>This function computes the probability 
<em>P(n_1, n_2, ..., n_K)</em>
of sampling <var>n</var>[<var>K</var>] from a multinomial distribution 
with parameters <var>p</var>[<var>K</var>], using the formula given above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fmultinomial_005flnpdf"></a>Function: <em>double</em> <strong>gsl_ran_multinomial_lnpdf</strong> <em>(size_t <var>K</var>, const double <var>p</var>[], const unsigned int <var>n</var>[]) </em></dt>
<dd><p>This function returns the logarithm of the probability for the
multinomial distribution <em>P(n_1, n_2, ..., n_K)</em> with parameters <var>p</var>[<var>K</var>].
</p></dd></dl>




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