.. index::

   single: random number distributions

   single: cumulative distribution functions (CDFs)

   single: CDFs, cumulative distribution functions

   single: inverse cumulative distribution functions

   single: quantile functions

.. _chap_random-number-distributions:

***************************

Random Number Distributions

***************************

.. include:: include.rst

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.  

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 *acceptance-rejection* method, which compares the desired

distribution against a distribution which is similar and known

analytically.  This usually requires several samples from the generator.

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.

The functions for random variates and probability density functions

described in this section are declared in :file:`gsl_randist.h`.  The

corresponding cumulative distribution functions are declared in

:file:`gsl_cdf.h`.

Note that the discrete random variate functions always

return a value of type :code:`unsigned int`, and on most platforms this

has a maximum value of

.. only:: not texinfo

   .. math:: 2^{32}-1 \approx 4.29 \times 10^9

.. only:: texinfo

   ::

      2^32-1 ~=~ 4.29e9

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.

Introduction

============

Continuous random number distributions are defined by a probability

density function, :math:`p(x)`, such that the probability of :math:`x`

occurring in the infinitesimal range :math:`x` to :math:`x + dx` is

:math:`p(x) dx`.

The cumulative distribution function for the lower tail :math:`P(x)` is

defined by the integral,

.. math:: P(x) = \int_{-\infty}^{x} dx' p(x')

and gives the probability of a variate taking a value less than :math:`x`.

The cumulative distribution function for the upper tail :math:`Q(x)` is

defined by the integral,

.. math:: Q(x) = \int_{x}^{+\infty} dx' p(x')

and gives the probability of a variate taking a value greater than :math:`x`.

The upper and lower cumulative distribution functions are related by

:math:`P(x) + Q(x) = 1` and satisfy :math:`0 \le P(x) \le 1`,

:math:`0 \le Q(x) \le 1`.

The inverse cumulative distributions, :math:`x = P^{-1}(P)`

and :math:`x = Q^{-1}(Q)`

give the values of :math:`x`

which correspond to a specific value of :math:`P` or :math:`Q`.  

They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer

value :math:`k` is given by :math:`p(k)`, where :math:`\sum_k p(k) = 1`.

The cumulative distribution for the lower tail :math:`P(k)` of a

discrete distribution is defined as,

.. math:: P(k) = \sum_{i \le k} p(i) 

where the sum is over the allowed range of the distribution less than

or equal to :math:`k`.  

The cumulative distribution for the upper tail of a discrete

distribution :math:`Q(k)` is defined as

.. math:: Q(k) = \sum_{i > k} p(i) 

giving the sum of probabilities for all values greater than :math:`k`.

These two definitions satisfy the identity :math:`P(k)+Q(k)=1`.

If the range of the distribution is 1 to :math:`n` inclusive then

:math:`P(n) = 1`, :math:`Q(n) = 0` while :math:`P(1) = p(1)`,

:math:`Q(1) = 1 - p(1)`.

|newpage|

The Gaussian Distribution

=========================

.. index:: Gaussian distribution

.. function:: double gsl_ran_gaussian (const gsl_rng * r, double sigma)

   This function returns a Gaussian random variate, with mean zero and

   standard deviation :data:`sigma`.  The probability distribution for

   Gaussian random variates is,

   .. math:: p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

   for :math:`x` in the range :math:`-\infty` to :math:`+\infty`.  Use the

   transformation :math:`z = \mu + x` on the numbers returned by

   :func:`gsl_ran_gaussian` to obtain a Gaussian distribution with mean

   :math:`\mu`.  This function uses the Box-Muller algorithm which requires two

   calls to the random number generator :data:`r`.

.. function:: double gsl_ran_gaussian_pdf (double x, double sigma)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Gaussian distribution with standard deviation :data:`sigma`, using

   the formula given above.

   .. image:: /images/rand-gaussian.png

.. index:: Ziggurat method

.. function:: double gsl_ran_gaussian_ziggurat (const gsl_rng * r, double sigma)

              double gsl_ran_gaussian_ratio_method (const gsl_rng * r, double sigma)

   This function computes a Gaussian random variate using the alternative

   Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods.  The

   Ziggurat algorithm is the fastest available algorithm in most cases.

.. function:: double gsl_ran_ugaussian (const gsl_rng * r)

              double gsl_ran_ugaussian_pdf (double x)

              double gsl_ran_ugaussian_ratio_method (const gsl_rng * r)

   These functions compute results for the unit Gaussian distribution.  They

   are equivalent to the functions above with a standard deviation of one,

   :data:`sigma` = 1.

.. function:: double gsl_cdf_gaussian_P (double x, double sigma)

              double gsl_cdf_gaussian_Q (double x, double sigma)

              double gsl_cdf_gaussian_Pinv (double P, double sigma)

              double gsl_cdf_gaussian_Qinv (double Q, double sigma)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the Gaussian

   distribution with standard deviation :data:`sigma`.

.. function:: double gsl_cdf_ugaussian_P (double x)

              double gsl_cdf_ugaussian_Q (double x)

              double gsl_cdf_ugaussian_Pinv (double P)

              double gsl_cdf_ugaussian_Qinv (double Q)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the unit Gaussian

   distribution.

|newpage|

The Gaussian Tail Distribution

==============================

.. index:: Gaussian Tail distribution

.. function:: double gsl_ran_gaussian_tail (const gsl_rng * r, double a, double sigma)

   This function provides random variates from the upper tail of a Gaussian

   distribution with standard deviation :data:`sigma`.  The values returned

   are larger than the lower limit :data:`a`, which must be positive.  The

   method is based on Marsaglia's famous rectangle-wedge-tail algorithm (Ann. 

   Math. Stat. 32, 894--899 (1961)), with this aspect explained in Knuth, v2,

   3rd ed, p139,586 (exercise 11).

   The probability distribution for Gaussian tail random variates is,

   .. math:: p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2 / 2\sigma^2) dx

   for :math:`x > a` where :math:`N(a;\sigma)` is the normalization constant,

   .. only:: not texinfo

      .. math:: N(a;\sigma) = {1 \over 2} \hbox{erfc}\left({a \over \sqrt{2 \sigma^2}}\right).

   .. only:: texinfo

      ::

         N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).

.. function:: double gsl_ran_gaussian_tail_pdf (double x, double a, double sigma)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Gaussian tail distribution with standard deviation :data:`sigma` and

   lower limit :data:`a`, using the formula given above.

   .. image:: /images/rand-gaussian-tail.png

.. function:: double gsl_ran_ugaussian_tail (const gsl_rng * r, double a)

              double gsl_ran_ugaussian_tail_pdf (double :data:`x`, double :data:`a`)

   These functions compute results for the tail of a unit Gaussian

   distribution.  They are equivalent to the functions above with a standard

   deviation of one, :data:`sigma` = 1.

|newpage|

The Bivariate Gaussian Distribution

===================================

.. index::

   single: Bivariate Gaussian distribution

   single: two dimensional Gaussian distribution

   single: Gaussian distribution, bivariate

.. function:: void gsl_ran_bivariate_gaussian (const gsl_rng * r, double sigma_x, double sigma_y, double rho, double * x, double * y)

   This function generates a pair of correlated Gaussian variates, with

   mean zero, correlation coefficient :data:`rho` and standard deviations

   :data:`sigma_x` and :data:`sigma_y` in the :math:`x` and :math:`y` directions.

   The probability distribution for bivariate Gaussian random variates is,

   .. only:: not texinfo

      .. math:: p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left(-{(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y)) \over 2(1-\rho^2)}\right) dx dy

   .. only:: texinfo

      ::

         p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

   for :math:`x,y` in the range :math:`-\infty` to :math:`+\infty`.  The

   correlation coefficient :data:`rho` should lie between :math:`1` and

   :math:`-1`.

.. function:: double gsl_ran_bivariate_gaussian_pdf (double x, double y, double sigma_x, double sigma_y, double rho)

   This function computes the probability density :math:`p(x,y)` at

   (:data:`x`, :data:`y`) for a bivariate Gaussian distribution with standard

   deviations :data:`sigma_x`, :data:`sigma_y` and correlation coefficient

   :data:`rho`, using the formula given above.

   .. image:: /images/rand-bivariate-gaussian.png

|newpage|

The Multivariate Gaussian Distribution

======================================

.. index::

   single: Bivariate Gaussian distribution

   single: two dimensional Gaussian distribution

   single: Gaussian distribution, bivariate

.. function:: int gsl_ran_multivariate_gaussian (const gsl_rng * r, const gsl_vector * mu, const gsl_matrix * L, gsl_vector * result)

   This function generates a random vector satisfying the :math:`k`-dimensional multivariate Gaussian

   distribution with mean :math:`\mu` and variance-covariance matrix

   :math:`\Sigma`. On input, the :math:`k`-vector :math:`\mu` is given in :data:`mu`, and

   the Cholesky factor of the :math:`k`-by-:math:`k` matrix :math:`\Sigma = L L^T` is

   given in the lower triangle of :data:`L`, as output from :func:`gsl_linalg_cholesky_decomp`.

   The random vector is stored in :data:`result` on output. The probability distribution

   for multivariate Gaussian random variates is

   .. only:: not texinfo

      .. math:: p(x_1,\dots,x_k) dx_1 \dots dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|}} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k

   .. only:: texinfo

      ::

         p(x_1,...,x_k) dx_1 ... dx_k = 1 / ( \sqrt{(2 \pi)^k |\Sigma| ) \exp (-1/2 (x - \mu)^T \Sigma^{-1} (x - \mu)) dx_1 ... dx_k

.. function:: int gsl_ran_multivariate_gaussian_pdf (const gsl_vector * x, const gsl_vector * mu, const gsl_matrix * L, double * result, gsl_vector * work)

              int gsl_ran_multivariate_gaussian_log_pdf (const gsl_vector * x, const gsl_vector * mu, const gsl_matrix * L, double * result, gsl_vector * work)

   These functions compute :math:`p(x)` or :math:`\log{p(x)}` at the point :data:`x`, using mean vector

   :data:`mu` and variance-covariance matrix specified by its Cholesky factor :data:`L` using the formula

   above. Additional workspace of length :math:`k` is required in :data:`work`.

.. function:: int gsl_ran_multivariate_gaussian_mean (const gsl_matrix * X, gsl_vector * mu_hat)

   Given a set of :math:`n` samples :math:`X_j` from a :math:`k`-dimensional multivariate Gaussian distribution,

   this function computes the maximum likelihood estimate of the mean of the distribution, given by

   .. math:: \Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j

   The samples :math:`X_1,X_2,\dots,X_n` are given in the :math:`n`-by-:math:`k` matrix :data:`X`, and the maximum

   likelihood estimate of the mean is stored in :data:`mu_hat` on output.

.. function:: int gsl_ran_multivariate_gaussian_vcov (const gsl_matrix * X, gsl_matrix * sigma_hat)

   Given a set of :math:`n` samples :math:`X_j` from a :math:`k`-dimensional multivariate Gaussian distribution,

   this function computes the maximum likelihood estimate of the variance-covariance matrix of the distribution,

   given by

   .. only:: not texinfo

      .. math:: \Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T

   .. only:: texinfo

      ::

         \Hat{\Sigma} = (1 / n) \sum_{j=1}^n ( X_j - \Hat{\mu} ) ( X_j - \Hat{\mu} )^T

   The samples :math:`X_1,X_2,\dots,X_n` are given in the :math:`n`-by-:math:`k` matrix :data:`X` and the maximum

   likelihood estimate of the variance-covariance matrix is stored in :data:`sigma_hat` on output.

|newpage|

The Exponential Distribution

============================

.. index:: Exponential distribution

.. function:: double gsl_ran_exponential (const gsl_rng * r, double mu)

   This function returns a random variate from the exponential distribution

   with mean :data:`mu`. The distribution is,

   .. math:: p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

   for :math:`x \ge 0`.

.. function:: double gsl_ran_exponential_pdf (double x, double mu)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for an exponential distribution with mean :data:`mu`, using the formula

   given above.

   .. image:: /images/rand-exponential.png

.. function:: double gsl_cdf_exponential_P (double x, double mu)

              double gsl_cdf_exponential_Q (double x, double mu)

              double gsl_cdf_exponential_Pinv (double P, double mu)

              double gsl_cdf_exponential_Qinv (double Q, double mu)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the exponential

   distribution with mean :data:`mu`.

|newpage|

The Laplace Distribution

========================

.. index::

   single: two-sided exponential distribution

   single: Laplace distribution

.. function:: double gsl_ran_laplace (const gsl_rng * r, double a)

   This function returns a random variate from the Laplace distribution

   with width :data:`a`.  The distribution is,

   .. math:: p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx

   for :math:`-\infty < x < \infty`.

.. function:: double gsl_ran_laplace_pdf (double x, double a)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Laplace distribution with width :data:`a`, using the formula

   given above.

   .. image:: /images/rand-laplace.png

.. function:: double gsl_cdf_laplace_P (double x, double a)

              double gsl_cdf_laplace_Q (double x, double a)

              double gsl_cdf_laplace_Pinv (double P, double a)

              double gsl_cdf_laplace_Qinv (double Q, double a)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the Laplace

   distribution with width :data:`a`.

|newpage|

The Exponential Power Distribution

==================================

.. index:: Exponential power distribution

.. function:: double gsl_ran_exppow (const gsl_rng * r, double a, double b)

   This function returns a random variate from the exponential power distribution

   with scale parameter :data:`a` and exponent :data:`b`.  The distribution is,

   .. math:: p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

   for :math:`x \ge 0`.

   For :math:`b = 1` this reduces to the Laplace

   distribution.  For :math:`b = 2` it has the same form as a Gaussian

   distribution, but with :math:`a = \sqrt{2} \sigma`.

.. function:: double gsl_ran_exppow_pdf (double x, double a, double b)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for an exponential power distribution with scale parameter :data:`a`

   and exponent :data:`b`, using the formula given above.

   .. image:: /images/rand-exppow.png

.. function:: double gsl_cdf_exppow_P (double x, double a, double b)

              double gsl_cdf_exppow_Q (double x, double a, double b)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` for the exponential power distribution with

   parameters :data:`a` and :data:`b`.

|newpage|

The Cauchy Distribution

=======================

.. index:: Cauchy distribution

.. function:: double gsl_ran_cauchy (const gsl_rng * r, double a)

   This function returns a random variate from the Cauchy distribution with

   scale parameter :data:`a`.  The probability distribution for Cauchy

   random variates is,

   .. math:: p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

   for :math:`x` in the range :math:`-\infty` to :math:`+\infty`.  The Cauchy

   distribution is also known as the Lorentz distribution.

.. function:: double gsl_ran_cauchy_pdf (double x, double a)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Cauchy distribution with scale parameter :data:`a`, using the formula

   given above.

   .. image:: /images/rand-cauchy.png

.. function:: double gsl_cdf_cauchy_P (double x, double a)

              double gsl_cdf_cauchy_Q (double x, double a)

              double gsl_cdf_cauchy_Pinv (double P, double a)

              double gsl_cdf_cauchy_Qinv (double Q, double a)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the Cauchy

   distribution with scale parameter :data:`a`.

|newpage|

The Rayleigh Distribution

=========================

.. index:: Rayleigh distribution

.. function:: double gsl_ran_rayleigh (const gsl_rng * r, double sigma)

   This function returns a random variate from the Rayleigh distribution with

   scale parameter :data:`sigma`.  The distribution is,

   .. math:: p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

   for :math:`x > 0`.

.. function:: double gsl_ran_rayleigh_pdf (double x, double sigma)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Rayleigh distribution with scale parameter :data:`sigma`, using the

   formula given above.

   .. image:: /images/rand-rayleigh.png

.. function:: double gsl_cdf_rayleigh_P (double x, double sigma)

              double gsl_cdf_rayleigh_Q (double x, double sigma)

              double gsl_cdf_rayleigh_Pinv (double P, double sigma)

              double gsl_cdf_rayleigh_Qinv (double Q, double sigma)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the Rayleigh

   distribution with scale parameter :data:`sigma`.

|newpage|

The Rayleigh Tail Distribution

==============================

.. index:: Rayleigh Tail distribution

.. function:: double gsl_ran_rayleigh_tail (const gsl_rng * r, double a, double sigma)

   This function returns a random variate from the tail of the Rayleigh

   distribution with scale parameter :data:`sigma` and a lower limit of

   :data:`a`.  The distribution is,

   .. math:: p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

   for :math:`x > a`.

.. function:: double gsl_ran_rayleigh_tail_pdf (double x, double a, double sigma)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a Rayleigh tail distribution with scale parameter :data:`sigma` and

   lower limit :data:`a`, using the formula given above.

   .. image:: /images/rand-rayleigh-tail.png

|newpage|

The Landau Distribution

=======================

.. index:: Landau distribution

.. function:: double gsl_ran_landau (const gsl_rng * r)

   This function returns a random variate from the Landau distribution.  The

   probability distribution for Landau random variates is defined

   analytically by the complex integral,

   .. only:: not texinfo

      .. math:: p(x) = {1 \over {2 \pi i}} \int_{c-i\infty}^{c+i\infty} ds\, \exp(s \log(s) + x s) 

   .. only:: texinfo

      ::

         p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s) 

   For numerical purposes it is more convenient to use the following

   equivalent form of the integral,

   .. math:: p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).

.. function:: double gsl_ran_landau_pdf (double x)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for the Landau distribution using an approximation to the formula given

   above.

   .. image:: /images/rand-landau.png

|newpage|

The Levy alpha-Stable Distributions

===================================

.. index:: Levy distribution

.. function:: double gsl_ran_levy (const gsl_rng * r, double c, double alpha)

   This function returns a random variate from the Levy symmetric stable

   distribution with scale :data:`c` and exponent :data:`alpha`.  The symmetric

   stable probability distribution is defined by a Fourier transform,

   .. only:: not texinfo

      .. math:: p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha)

   .. only:: texinfo

      ::

         p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

   There is no explicit solution for the form of :math:`p(x)` and the

   library does not define a corresponding :code:`pdf` function.  For

   :math:`\alpha = 1` the distribution reduces to the Cauchy distribution.  For

   :math:`\alpha = 2` it is a Gaussian distribution with :math:`\sigma = \sqrt{2} c`.

   For :math:`\alpha < 1` the tails of the distribution become extremely wide.

   The algorithm only works for :math:`0 < \alpha \le 2`.

   .. image:: /images/rand-levy.png

|newpage|

The Levy skew alpha-Stable Distribution

=======================================

.. index::

   single: Levy distribution, skew

   single: Skew Levy distribution

.. function:: double gsl_ran_levy_skew (const gsl_rng * r, double c, double alpha, double beta)

   This function returns a random variate from the Levy skew stable

   distribution with scale :data:`c`, exponent :data:`alpha` and skewness

   parameter :data:`beta`.  The skewness parameter must lie in the range

   :math:`[-1,1]`.  The Levy skew stable probability distribution is defined

   by a Fourier transform,

   .. only:: not texinfo

      .. math:: p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha (1-i \beta \sgn(t) \tan(\pi\alpha/2)))

   .. only:: texinfo

      ::

         p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

   When :math:`\alpha = 1` the term :math:`\tan(\pi \alpha/2)` is replaced by

   :math:`-(2/\pi)\log|t|`.  There is no explicit solution for the form of

   :math:`p(x)` and the library does not define a corresponding :code:`pdf`

   function.  For :math:`\alpha = 2` the distribution reduces to a Gaussian

   distribution with :math:`\sigma = \sqrt{2} c`

   and the skewness parameter has no effect.  

   For :math:`\alpha < 1` the tails of the distribution become extremely

   wide.  The symmetric distribution corresponds to :math:`\beta = 0`.

   The algorithm only works for :math:`0 < \alpha \le 2`.

The Levy alpha-stable distributions have the property that if :math:`N`

alpha-stable variates are drawn from the distribution :math:`p(c, \alpha, \beta)`

then the sum :math:`Y = X_1 + X_2 + \dots + X_N` will also be

distributed as an alpha-stable variate,

:math:`p(N^{1/\alpha} c, \alpha, \beta)`.

.. PDF not available because there is no analytic expression for it

..

.. @deftypefun double gsl_ran_levy_pdf (double x, double mu)

.. This function computes the probability density :math:`p(x)` at :data:`x`

.. for a symmetric Levy distribution with scale parameter :data:`mu` and

.. exponent :data:`a`, using the formula given above.

.. @end deftypefun

.. image:: /images/rand-levyskew.png

|newpage|

The Gamma Distribution

======================

.. index::

   single: Gamma distribution

   single: Erlang distribution

.. function:: double gsl_ran_gamma (const gsl_rng * r, double a, double b)

   This function returns a random variate from the gamma

   distribution.  The distribution function is,

   .. math:: p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

   for :math:`x > 0`.

   The gamma distribution with an integer parameter :data:`a` is known as the Erlang distribution.

   The variates are computed using the Marsaglia-Tsang fast gamma method.

   This function for this method was previously called

   :func:`gsl_ran_gamma_mt` and can still be accessed using this name.

.. If @xmath{X} and @xmath{Y} are independent gamma-distributed random

.. variables of order @xmath{a} and @xmath{b}, then @xmath{X+Y} has a gamma

.. distribution of order @xmath{a+b}.

.. function:: double gsl_ran_gamma_knuth (const gsl_rng * r, double a, double b)

   This function returns a gamma variate using the algorithms from Knuth (vol 2).

.. function:: double gsl_ran_gamma_pdf (double x, double a, double b)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a gamma distribution with parameters :data:`a` and :data:`b`, using the

   formula given above.

   .. image:: /images/rand-gamma.png

.. function:: double gsl_cdf_gamma_P (double x, double a, double b)

              double gsl_cdf_gamma_Q (double x, double a, double b)

              double gsl_cdf_gamma_Pinv (double P, double a, double b)

              double gsl_cdf_gamma_Qinv (double Q, double a, double b)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the gamma

   distribution with parameters :data:`a` and :data:`b`.

|newpage|

The Flat (Uniform) Distribution

===============================

.. index::

   single: flat distribution

   single: uniform distribution

.. function:: double gsl_ran_flat (const gsl_rng * r, double a, double b)

   This function returns a random variate from the flat (uniform)

   distribution from :data:`a` to :data:`b`. The distribution is,

   .. math:: p(x) dx = {1 \over (b-a)} dx

   if :math:`a \le x < b` and 0 otherwise.

.. function:: double gsl_ran_flat_pdf (double x, double a, double b)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a uniform distribution from :data:`a` to :data:`b`, using the formula

   given above.

   .. image:: /images/rand-flat.png

.. function:: double gsl_cdf_flat_P (double x, double a, double b)

              double gsl_cdf_flat_Q (double x, double a, double b)

              double gsl_cdf_flat_Pinv (double P, double a, double b)

              double gsl_cdf_flat_Qinv (double Q, double a, double b)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for a uniform distribution

   from :data:`a` to :data:`b`.

|newpage|

The Lognormal Distribution

==========================

.. index:: Lognormal distribution

.. function:: double gsl_ran_lognormal (const gsl_rng * r, double zeta, double sigma)

   This function returns a random variate from the lognormal

   distribution.  The distribution function is,

   .. math:: p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

   for :math:`x > 0`.

.. function:: double gsl_ran_lognormal_pdf (double x, double zeta, double sigma)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a lognormal distribution with parameters :data:`zeta` and :data:`sigma`,

   using the formula given above.

   .. image:: /images/rand-lognormal.png

.. function:: double gsl_cdf_lognormal_P (double x, double zeta, double sigma)

              double gsl_cdf_lognormal_Q (double x, double zeta, double sigma)

              double gsl_cdf_lognormal_Pinv (double P, double zeta, double sigma)

              double gsl_cdf_lognormal_Qinv (double Q, double zeta, double sigma)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the lognormal

   distribution with parameters :data:`zeta` and :data:`sigma`.

|newpage|

The Chi-squared Distribution

============================

The chi-squared distribution arises in statistics.  If :math:`Y_i` are

:math:`n` independent Gaussian random variates with unit variance then the

sum-of-squares,

.. math:: X_i = \sum_i Y_i^2

has a chi-squared distribution with :math:`n` degrees of freedom.

.. index:: Chi-squared distribution

.. function:: double gsl_ran_chisq (const gsl_rng * r, double nu)

   This function returns a random variate from the chi-squared distribution

   with :data:`nu` degrees of freedom. The distribution function is,

   .. math:: p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

   for :math:`x \ge 0`.

.. function:: double gsl_ran_chisq_pdf (double x, double nu)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for a chi-squared distribution with :data:`nu` degrees of freedom, using

   the formula given above.

   .. image:: /images/rand-chisq.png

.. function:: double gsl_cdf_chisq_P (double x, double nu)

              double gsl_cdf_chisq_Q (double x, double nu)

              double gsl_cdf_chisq_Pinv (double P, double nu)

              double gsl_cdf_chisq_Qinv (double Q, double nu)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the chi-squared

   distribution with :data:`nu` degrees of freedom.

|newpage|

The F-distribution

==================

The F-distribution arises in statistics.  If :math:`Y_1` and :math:`Y_2`

are chi-squared deviates with :math:`\nu_1` and :math:`\nu_2` degrees of

freedom then the ratio,

.. math:: X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

has an F-distribution :math:`F(x;\nu_1,\nu_2)`.

.. index:: F-distribution

.. function:: double gsl_ran_fdist (const gsl_rng * r, double nu1, double nu2)

   This function returns a random variate from the F-distribution with degrees of freedom :data:`nu1` and :data:`nu2`.

   The distribution function is,

   .. only:: not texinfo

      .. math::

         p(x) dx = 

            { \Gamma((\nu_1 + \nu_2)/2)

                 \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 

            \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 

            x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

   .. only:: texinfo

      ::

         p(x) dx = 

            { \Gamma((\nu_1 + \nu_2)/2)

                 \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 

            \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 

            x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

   for :math:`x \ge 0`.

.. function:: double gsl_ran_fdist_pdf (double x, double nu1, double nu2)

   This function computes the probability density :math:`p(x)` at :data:`x`

   for an F-distribution with :data:`nu1` and :data:`nu2` degrees of freedom,

   using the formula given above.

   .. image:: /images/rand-fdist.png

.. function:: double gsl_cdf_fdist_P (double x, double nu1, double nu2)

              double gsl_cdf_fdist_Q (double x, double nu1, double nu2)

              double gsl_cdf_fdist_Pinv (double P, double nu1, double nu2)

              double gsl_cdf_fdist_Qinv (double Q, double nu1, double nu2)

   These functions compute the cumulative distribution functions

   :math:`P(x)`, :math:`Q(x)` and their inverses for the F-distribution

   with :data:`nu1` and :data:`nu2` degrees of freedom.

|newpage|

The t-distribution

==================

The t-distribution arises in statistics.  If :math:`Y_1` has a normal

distribution and :math:`Y_2` has a chi-squared distribution with

:math:`\nu` degrees of freedom then the ratio,

.. math:: X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution :math:`t(x;\nu)` with :math:`\nu` degrees of freedom.

.. index::

   single: t-distribution