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<title>The Multinomial Distribution - GNU Scientific Library -- Reference Manual</title>
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<p>
<a name="The-Multinomial-Distribution"></a>
Next: <a rel="next" accesskey="n" href="The-Negative-Binomial-Distribution.html">The Negative Binomial Distribution</a>,
Previous: <a rel="previous" accesskey="p" href="The-Binomial-Distribution.html">The Binomial Distribution</a>,
Up: <a rel="up" accesskey="u" href="Random-Number-Distributions.html">Random Number Distributions</a>
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<h3 class="section">19.32 The Multinomial Distribution</h3>
<div class="defun">
— Function: void <b>gsl_ran_multinomial</b> (<var>const gsl_rng * r, size_t K, unsigned int N, const double p</var>[]<var>, unsigned int n</var>[])<var><a name="index-gsl_005fran_005fmultinomial-1778"></a></var><br>
<blockquote><p><a name="index-Multinomial-distribution-1779"></a>
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,
<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>
<p class="noindent">where <!-- {($n_1$, $n_2$, $\ldots$, $n_K$)} -->
(n_1, n_2, ..., n_K)
are nonnegative integers with
<!-- {$\sum_{k=1}^{K} n_k =N$} -->
sum_{k=1}^K n_k = N,
and
<!-- {$(p_1, p_2, \ldots, p_K)$} -->
(p_1, p_2, ..., p_K)
is a probability distribution with \sum p_i = 1.
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>Random variates are generated using the conditional binomial method (see
C.S. David, <cite>The computer generation of multinomial random
variates</cite>, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_ran_multinomial_pdf</b> (<var>size_t K, const double p</var>[]<var>, const unsigned int n</var>[])<var><a name="index-gsl_005fran_005fmultinomial_005fpdf-1780"></a></var><br>
<blockquote><p>This function computes the probability
<!-- {$P(n_1, n_2, \ldots, n_K)$} -->
P(n_1, n_2, ..., n_K)
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></blockquote></div>
<div class="defun">
— Function: double <b>gsl_ran_multinomial_lnpdf</b> (<var>size_t K, const double p</var>[]<var>, const unsigned int n</var>[])<var><a name="index-gsl_005fran_005fmultinomial_005flnpdf-1781"></a></var><br>
<blockquote><p>This function returns the logarithm of the probability for the
multinomial distribution <!-- {$P(n_1, n_2, \ldots, n_K)$} -->
P(n_1, n_2, ..., n_K) with parameters <var>p</var>[<var>K</var>].
</p></blockquote></div>
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