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<title>GNU Scientific Library &ndash; Reference Manual: Random Number Distributions</title>

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<a name="Random-Number-Distributions"></a>
<div class="header">
<p>
Next: <a href="Statistics.html#Statistics" accesskey="n" rel="next">Statistics</a>, Previous: <a href="Quasi_002dRandom-Sequences.html#Quasi_002dRandom-Sequences" accesskey="p" rel="previous">Quasi-Random Sequences</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Random-Number-Distributions-1"></a>
<h2 class="chapter">20 Random Number Distributions</h2>
<a name="index-random-number-distributions"></a>
<a name="index-cumulative-distribution-functions-_0028CDFs_0029"></a>
<a name="index-CDFs_002c-cumulative-distribution-functions"></a>
<a name="index-inverse-cumulative-distribution-functions"></a>
<a name="index-quantile-functions"></a>
<p>This chapter describes functions for generating random variates and
computing their probability distributions.  Samples from the
distributions described in this chapter can be obtained using any of the
random number generators in the library as an underlying source of
randomness.  
</p>
<p>In the simplest cases a non-uniform distribution can be obtained
analytically from the uniform distribution of a random number generator
by applying an appropriate transformation.  This method uses one call to
the random number generator.  More complicated distributions are created
by the <em>acceptance-rejection</em> method, which compares the desired
distribution against a distribution which is similar and known
analytically.  This usually requires several samples from the generator.
</p>
<p>The library also provides cumulative distribution functions and inverse
cumulative distribution functions, sometimes referred to as quantile
functions.  The cumulative distribution functions and their inverses are
computed separately for the upper and lower tails of the distribution,
allowing full accuracy to be retained for small results.
</p>
<p>The functions for random variates and probability density functions
described in this section are declared in <samp>gsl_randist.h</samp>.  The
corresponding cumulative distribution functions are declared in
<samp>gsl_cdf.h</samp>.
</p>
<p>Note that the discrete random variate functions always
return a value of type <code>unsigned int</code>, and on most platforms this
has a maximum value of <em>2^32-1 ~=~ 4.29e9</em>. They should only be called with
a safe range of parameters (where there is a negligible probability of
a variate exceeding this limit) to prevent incorrect results due to
overflow.
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="Random-Number-Distribution-Introduction.html#Random-Number-Distribution-Introduction" accesskey="1">Random Number Distribution Introduction</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Gaussian-Distribution.html#The-Gaussian-Distribution" accesskey="2">The Gaussian Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Gaussian-Tail-Distribution.html#The-Gaussian-Tail-Distribution" accesskey="3">The Gaussian Tail Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Bivariate-Gaussian-Distribution.html#The-Bivariate-Gaussian-Distribution" accesskey="4">The Bivariate Gaussian Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Multivariate-Gaussian-Distribution.html#The-Multivariate-Gaussian-Distribution" accesskey="5">The Multivariate Gaussian Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Exponential-Distribution.html#The-Exponential-Distribution" accesskey="6">The Exponential Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Laplace-Distribution.html#The-Laplace-Distribution" accesskey="7">The Laplace Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Exponential-Power-Distribution.html#The-Exponential-Power-Distribution" accesskey="8">The Exponential Power Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Cauchy-Distribution.html#The-Cauchy-Distribution" accesskey="9">The Cauchy Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Rayleigh-Distribution.html#The-Rayleigh-Distribution">The Rayleigh Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Rayleigh-Tail-Distribution.html#The-Rayleigh-Tail-Distribution">The Rayleigh Tail Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Landau-Distribution.html#The-Landau-Distribution">The Landau Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Levy-alpha_002dStable-Distributions.html#The-Levy-alpha_002dStable-Distributions">The Levy alpha-Stable Distributions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Levy-skew-alpha_002dStable-Distribution.html#The-Levy-skew-alpha_002dStable-Distribution">The Levy skew alpha-Stable Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Gamma-Distribution.html#The-Gamma-Distribution">The Gamma Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Flat-_0028Uniform_0029-Distribution.html#The-Flat-_0028Uniform_0029-Distribution">The Flat (Uniform) Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Lognormal-Distribution.html#The-Lognormal-Distribution">The Lognormal Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Chi_002dsquared-Distribution.html#The-Chi_002dsquared-Distribution">The Chi-squared Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-F_002ddistribution.html#The-F_002ddistribution">The F-distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-t_002ddistribution.html#The-t_002ddistribution">The t-distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Beta-Distribution.html#The-Beta-Distribution">The Beta Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Logistic-Distribution.html#The-Logistic-Distribution">The Logistic Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Pareto-Distribution.html#The-Pareto-Distribution">The Pareto Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Spherical-Vector-Distributions.html#Spherical-Vector-Distributions">Spherical Vector Distributions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Weibull-Distribution.html#The-Weibull-Distribution">The Weibull Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Type_002d1-Gumbel-Distribution.html#The-Type_002d1-Gumbel-Distribution">The Type-1 Gumbel Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Type_002d2-Gumbel-Distribution.html#The-Type_002d2-Gumbel-Distribution">The Type-2 Gumbel Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Dirichlet-Distribution.html#The-Dirichlet-Distribution">The Dirichlet Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="General-Discrete-Distributions.html#General-Discrete-Distributions">General Discrete Distributions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Poisson-Distribution.html#The-Poisson-Distribution">The Poisson Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Bernoulli-Distribution.html#The-Bernoulli-Distribution">The Bernoulli Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Binomial-Distribution.html#The-Binomial-Distribution">The Binomial Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Multinomial-Distribution.html#The-Multinomial-Distribution">The Multinomial Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Negative-Binomial-Distribution.html#The-Negative-Binomial-Distribution">The Negative Binomial Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Pascal-Distribution.html#The-Pascal-Distribution">The Pascal Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Geometric-Distribution.html#The-Geometric-Distribution">The Geometric Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Hypergeometric-Distribution.html#The-Hypergeometric-Distribution">The Hypergeometric Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-Logarithmic-Distribution.html#The-Logarithmic-Distribution">The Logarithmic Distribution</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Shuffling-and-Sampling.html#Shuffling-and-Sampling">Shuffling and Sampling</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Random-Number-Distribution-Examples.html#Random-Number-Distribution-Examples">Random Number Distribution Examples</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Random-Number-Distribution-References-and-Further-Reading.html#Random-Number-Distribution-References-and-Further-Reading">Random Number Distribution References and Further Reading</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<hr>
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<p>
Next: <a href="Statistics.html#Statistics" accesskey="n" rel="next">Statistics</a>, Previous: <a href="Quasi_002dRandom-Sequences.html#Quasi_002dRandom-Sequences" accesskey="p" rel="previous">Quasi-Random Sequences</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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