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 gsl_randist.h.  The corresponding cumulative distribution functions are declared in gsl_cdf.h.

Note that the discrete random variate functions always return a value of type unsigned int, and on most platforms this has a maximum value of $2^32-1 \approx 4.29\times10^9$ 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, p(x), such that the probability of x occurring in the infinitesimal range x to x+dx is $p\,dx$ p dx.

The cumulative distribution function for the lower tail P(x) is defined by the integral,

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

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

The cumulative distribution function for the upper tail Q(x) is defined by the integral,

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

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

The upper and lower cumulative distribution functions are related by P(x) + Q(x) = 1 and satisfy $0 \le P(x) \le 1$ 0 <= P(x) <= 1, $0 \le Q(x) \le 1$ 0 <= Q(x) <= 1.

The inverse cumulative distributions, $x=P^-1(P)$ x=P^{-1}(P) and $x=Q^-1(Q)$ x=Q^{-1}(Q) give the values of x which correspond to a specific value of P or Q. They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value k is given by p(k), where \sum_k p(k) = 1. The cumulative distribution for the lower tail P(k) of a discrete distribution is defined as,

     P(k) = \sum_{i <= k} p(i)

where the sum is over the allowed range of the distribution less than or equal to k.

The cumulative distribution for the upper tail of a discrete distribution Q(k) is defined as

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

giving the sum of probabilities for all values greater than k. These two definitions satisfy the identity P(k)+Q(k)=1.

If the range of the distribution is 1 to n inclusive then P(n)=1, Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).

The 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 SIGMA.  The probability distribution for Gaussian random variates is,

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

     for x in the range -\infty to +\infty.  Use the transformation z = \mu + x on the numbers returned by gsl_ran_gaussian to obtain a Gaussian distribution with mean \mu.  This function uses the Box-Mueller algorithm which requires two calls to the random number generator R.

 -- Function: double gsl_ran_gaussian_pdf (double X, double SIGMA)
     This function computes the probability density p(x) at X for a Gaussian distribution with standard deviation SIGMA, using the formula given above.

 -- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng * R, double SIGMA)
 -- Function: 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)
 -- Function: double gsl_ran_ugaussian_pdf (double X)
 -- Function: 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, SIGMA = 1.

 -- Function: double gsl_cdf_gaussian_P (double X, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Q (double X, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Pinv (double P, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Qinv (double Q, double SIGMA)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Gaussian distribution with standard deviation SIGMA.

 -- Function: double gsl_cdf_ugaussian_P (double X)
 -- Function: double gsl_cdf_ugaussian_Q (double X)
 -- Function: double gsl_cdf_ugaussian_Pinv (double P)
 -- Function: double gsl_cdf_ugaussian_Qinv (double Q)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the unit Gaussian distribution.

The 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 SIGMA.  The values returned are larger than the lower limit 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,

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

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

          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 p(x) at X for a Gaussian tail distribution with standard deviation SIGMA and lower limit A, using the formula given above.

 -- Function: double gsl_ran_ugaussian_tail (const gsl_rng * R, double A)
 -- Function: double gsl_ran_ugaussian_tail_pdf (double X, double 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, SIGMA = 1.

The Bivariate Gaussian Distribution
===================================

 -- 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 RHO and standard deviations SIGMA_X and SIGMA_Y in the x and y directions. The probability distribution for bivariate Gaussian random variates is,

          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 x,y in the range -\infty to +\infty.  The correlation coefficient RHO should lie between 1 and -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 p(x,y) at (X,Y) for a bivariate Gaussian distribution with standard deviations SIGMA_X, SIGMA_Y and correlation coefficient RHO, using the formula given above.

The 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 MU. The distribution is,

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

     for $x \ge 0$ x >= 0.

 -- Function: double gsl_ran_exponential_pdf (double X, double MU)
     This function computes the probability density p(x) at X for an exponential distribution with mean MU, using the formula given above.

 -- Function: double gsl_cdf_exponential_P (double X, double MU)
 -- Function: double gsl_cdf_exponential_Q (double X, double MU)
 -- Function: double gsl_cdf_exponential_Pinv (double P, double MU)
 -- Function: double gsl_cdf_exponential_Qinv (double Q, double MU)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the exponential distribution with mean MU.

The 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 A.  The distribution is,

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

     for -\infty < x < \infty.

 -- Function: double gsl_ran_laplace_pdf (double X, double A)
     This function computes the probability density p(x) at X for a Laplace distribution with width A, using the formula given above.

 -- Function: double gsl_cdf_laplace_P (double X, double A)
 -- Function: double gsl_cdf_laplace_Q (double X, double A)
 -- Function: double gsl_cdf_laplace_Pinv (double P, double A)
 -- Function: double gsl_cdf_laplace_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Laplace distribution with width A.

The 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 A and exponent B.  The distribution is,

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

     for $x \ge 0$ x >= 0.  For b = 1 this reduces to the Laplace distribution.  For b = 2 it has the same form as a gaussian distribution, but with $a = \sqrt2 \sigma$ a = \sqrt{2} \sigma.

 -- Function: double gsl_ran_exppow_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for an exponential power distribution with scale parameter A and exponent B, using the formula given above.

 -- Function: double gsl_cdf_exppow_P (double X, double A, double B)
 -- Function: double gsl_cdf_exppow_Q (double X, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) for the exponential power distribution with parameters A and B.

The 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 A.  The probability distribution for Cauchy random variates is,

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

     for x in the range -\infty to +\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 p(x) at X for a Cauchy distribution with scale parameter A, using the formula given above.

 -- Function: double gsl_cdf_cauchy_P (double X, double A)
 -- Function: double gsl_cdf_cauchy_Q (double X, double A)
 -- Function: double gsl_cdf_cauchy_Pinv (double P, double A)
 -- Function: double gsl_cdf_cauchy_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Cauchy distribution with scale parameter A.

The 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 SIGMA.  The distribution is,

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

     for x > 0.

 -- Function: double gsl_ran_rayleigh_pdf (double X, double SIGMA)
     This function computes the probability density p(x) at X for a Rayleigh distribution with scale parameter SIGMA, using the formula given above.

 -- Function: double gsl_cdf_rayleigh_P (double X, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Q (double X, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Pinv (double P, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Qinv (double Q, double SIGMA)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Rayleigh distribution with scale parameter SIGMA.

The 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 SIGMA and a lower limit of A.  The distribution is,

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

     for x > a.

 -- Function: double gsl_ran_rayleigh_tail_pdf (double X, double A, double SIGMA)
     This function computes the probability density p(x) at X for a Rayleigh tail distribution with scale parameter SIGMA and lower limit A, using the formula given above.

The 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,

          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,

          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 p(x) at X for the Landau distribution using an approximation to the formula given above.

The Levy alpha-Stable Distributions
===================================

 -- 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 C and exponent ALPHA.  The symmetric stable probability distribution is defined by a fourier transform,

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

     There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function.  For \alpha = 1 the distribution reduces to the Cauchy distribution.  For \alpha = 2 it is a Gaussian distribution with $\sigma = \sqrt2 c$ \sigma = \sqrt{2} c.  For \alpha < 1 the tails of the distribution become extremely wide.

     The algorithm only works for $0 < \alpha \le 2$ 0 < alpha <= 2.

The Levy skew alpha-Stable 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 C, exponent ALPHA and skewness parameter BETA.  The skewness parameter must lie in the range [-1,1].  The Levy skew stable probability distribution is defined by a fourier transform,

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

     When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by -(2/\pi)\log|t|.  There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function.  For \alpha = 2 the distribution reduces to a Gaussian distribution with $\sigma = \sqrt2 c$ \sigma = \sqrt{2} c and the skewness parameter has no effect. For \alpha < 1 the tails of the distribution become extremely wide.  The symmetric distribution corresponds to \beta = 0.

     The algorithm only works for $0 < \alpha \le 2$ 0 < alpha <= 2.

The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \alpha, \beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed as an alpha-stable variate, $p(N^1/\alpha c, \alpha, \beta)$ p(N^(1/\alpha) c, \alpha, \beta).

The Gamma 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,

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

     for x > 0.

     The gamma distribution with an integer parameter 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 gsl_ran_gamma_mt and can still be accessed using this name.

 -- 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 p(x) at X for a gamma distribution with parameters A and B, using the formula given above.

 -- Function: double gsl_cdf_gamma_P (double X, double A, double B)
 -- Function: double gsl_cdf_gamma_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gamma_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gamma_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the gamma distribution with parameters A and B.

The Flat (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 A to B. The distribution is,

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

     if $a \le x < b$ a <= x < b and 0 otherwise.

 -- Function: double gsl_ran_flat_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a uniform distribution from A to B, using the formula given above.

 -- Function: double gsl_cdf_flat_P (double X, double A, double B)
 -- Function: double gsl_cdf_flat_Q (double X, double A, double B)
 -- Function: double gsl_cdf_flat_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_flat_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for a uniform distribution from A to B.

The 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,

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

     for x > 0.

 -- Function: double gsl_ran_lognormal_pdf (double X, double ZETA, double SIGMA)
     This function computes the probability density p(x) at X for a lognormal distribution with parameters ZETA and SIGMA, using the formula given above.

 -- Function: double gsl_cdf_lognormal_P (double X, double ZETA, double SIGMA)
 -- Function: double gsl_cdf_lognormal_Q (double X, double ZETA, double SIGMA)
 -- Function: double gsl_cdf_lognormal_Pinv (double P, double ZETA, double SIGMA)
 -- Function: double gsl_cdf_lognormal_Qinv (double Q, double ZETA, double SIGMA)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the lognormal distribution with parameters ZETA and SIGMA.

The Chi-squared Distribution
============================

The chi-squared distribution arises in statistics.  If Y_i are n independent gaussian random variates with unit variance then the sum-of-squares,

     X_i = \sum_i Y_i^2

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

 -- Function: double gsl_ran_chisq (const gsl_rng * R, double NU)
     This function returns a random variate from the chi-squared distribution with NU degrees of freedom. The distribution function is,

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

     for $x \ge 0$ x >= 0.

 -- Function: double gsl_ran_chisq_pdf (double X, double NU)
     This function computes the probability density p(x) at X for a chi-squared distribution with NU degrees of freedom, using the formula given above.

 -- Function: double gsl_cdf_chisq_P (double X, double NU)
 -- Function: double gsl_cdf_chisq_Q (double X, double NU)
 -- Function: double gsl_cdf_chisq_Pinv (double P, double NU)
 -- Function: double gsl_cdf_chisq_Qinv (double Q, double NU)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the chi-squared distribution with NU degrees of freedom.

The F-distribution
==================

The F-distribution arises in statistics.  If Y_1 and Y_2 are chi-squared deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,

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

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

 -- 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 NU1 and NU2. The distribution function is,

          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 $x \ge 0$ x >= 0.

 -- Function: double gsl_ran_fdist_pdf (double X, double NU1, double NU2)
     This function computes the probability density p(x) at X for an F-distribution with NU1 and NU2 degrees of freedom, using the formula given above.

 -- Function: double gsl_cdf_fdist_P (double X, double NU1, double NU2)
 -- Function: double gsl_cdf_fdist_Q (double X, double NU1, double NU2)
 -- Function: double gsl_cdf_fdist_Pinv (double P, double NU1, double NU2)
 -- Function: double gsl_cdf_fdist_Qinv (double Q, double NU1, double NU2)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the F-distribution with NU1 and NU2 degrees of freedom.

The t-distribution
==================

The t-distribution arises in statistics.  If Y_1 has a normal distribution and Y_2 has a chi-squared distribution with \nu degrees of freedom then the ratio,

     X = { Y_1 \over \sqrt{Y_2 / \nu} }

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

 -- Function: double gsl_ran_tdist (const gsl_rng * R, double NU)
     This function returns a random variate from the t-distribution.  The distribution function is,

          p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
             (1 + x^2/\nu)^{-(\nu + 1)/2} dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_tdist_pdf (double X, double NU)
     This function computes the probability density p(x) at X for a t-distribution with NU degrees of freedom, using the formula given above.

 -- Function: double gsl_cdf_tdist_P (double X, double NU)
 -- Function: double gsl_cdf_tdist_Q (double X, double NU)
 -- Function: double gsl_cdf_tdist_Pinv (double P, double NU)
 -- Function: double gsl_cdf_tdist_Qinv (double Q, double NU)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the t-distribution with NU degrees of freedom.

The Beta Distribution
=====================

 -- Function: double gsl_ran_beta (const gsl_rng * R, double A, double B)
     This function returns a random variate from the beta distribution.  The distribution function is,

          p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

     for $0 \le x \le 1$ 0 <= x <= 1.

 -- Function: double gsl_ran_beta_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a beta distribution with parameters A and B, using the formula given above.

 -- Function: double gsl_cdf_beta_P (double X, double A, double B)
 -- Function: double gsl_cdf_beta_Q (double X, double A, double B)
 -- Function: double gsl_cdf_beta_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_beta_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the beta distribution with parameters A and B.

The Logistic Distribution
=========================

 -- Function: double gsl_ran_logistic (const gsl_rng * R, double A)
     This function returns a random variate from the logistic distribution.  The distribution function is,

          p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_logistic_pdf (double X, double A)
     This function computes the probability density p(x) at X for a logistic distribution with scale parameter A, using the formula given above.

 -- Function: double gsl_cdf_logistic_P (double X, double A)
 -- Function: double gsl_cdf_logistic_Q (double X, double A)
 -- Function: double gsl_cdf_logistic_Pinv (double P, double A)
 -- Function: double gsl_cdf_logistic_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the logistic distribution with scale parameter A.

The Pareto Distribution
=======================

 -- Function: double gsl_ran_pareto (const gsl_rng * R, double A, double B)
     This function returns a random variate from the Pareto distribution of order A.  The distribution function is,

          p(x) dx = (a/b) / (x/b)^{a+1} dx

     for $x \ge b$ x >= b.

 -- Function: double gsl_ran_pareto_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a Pareto distribution with exponent A and scale B, using the formula given above.

 -- Function: double gsl_cdf_pareto_P (double X, double A, double B)
 -- Function: double gsl_cdf_pareto_Q (double X, double A, double B)
 -- Function: double gsl_cdf_pareto_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_pareto_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Pareto distribution with exponent A and scale B.

Spherical Vector Distributions
==============================

The spherical distributions generate random vectors, located on a spherical surface.  They can be used as random directions, for example in the steps of a random walk.

 -- Function: void gsl_ran_dir_2d (const gsl_rng * R, double * X, double * Y)
 -- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng * R, double * X, double * Y)
     This function returns a random direction vector v = (X,Y) in two dimensions.  The vector is normalized such that |v|^2 = x^2 + y^2 = 1.  The obvious way to do this is to take a uniform random number between 0 and 2\pi and let X and Y be the sine and cosine respectively.  Two trig functions would have been expensive in the old days, but with modern hardware implementations, this is sometimes the fastest way to go.  This is the case for the Pentium (but not the case for the Sun Sparcstation). One can avoid the trig evaluations by choosing X and Y in the interior of a unit circle (choose them at random from the interior of the enclosing square, and then reject those that are outside the unit circle), and then dividing by $\sqrtx^2 + y^2$ \sqrt{x^2 + y^2}. A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise 23), requires neither trig nor a square root.  In this approach, U and V are chosen at random from the interior of a unit circle, and then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).

 -- Function: void gsl_ran_dir_3d (const gsl_rng * R, double * X, double * Y, double * Z)
     This function returns a random direction vector v = (X,Y,Z) in three dimensions.  The vector is normalized such that |v|^2 = x^2 + y^2 + z^2 = 1.  The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136.  It uses the surprising fact that the distribution projected along any axis is actually uniform (this is only true for 3 dimensions).

 -- Function: void gsl_ran_dir_nd (const gsl_rng * R, size_t N, double * X)

     This function returns a random direction vector $v = (x_1,x_2,\ldots,x_n)$ v = (x_1,x_2,...,x_n) in N dimensions.  The vector is normalized such that $|v|^2 = x_1^2 + x_2^2 + \cdots + x_n^2 = 1$ |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1.  The method uses the fact that a multivariate gaussian distribution is spherically symmetric.  Each component is generated to have a gaussian distribution, and then the components are normalized.  The method is described by Knuth, v2, 3rd ed, p135--136, and attributed to G. W. Brown, Modern Mathematics for the Engineer (1956).

The Weibull Distribution
========================

 -- Function: double gsl_ran_weibull (const gsl_rng * R, double A, double B)
     This function returns a random variate from the Weibull distribution.  The distribution function is,

          p(x) dx = {b \over a^b} x^{b-1}  \exp(-(x/a)^b) dx

     for $x \ge 0$ x >= 0.

 -- Function: double gsl_ran_weibull_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a Weibull distribution with scale A and exponent B, using the formula given above.

 -- Function: double gsl_cdf_weibull_P (double X, double A, double B)
 -- Function: double gsl_cdf_weibull_Q (double X, double A, double B)
 -- Function: double gsl_cdf_weibull_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_weibull_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Weibull distribution with scale A and exponent B.

The Type-1 Gumbel Distribution
==============================

 -- Function: double gsl_ran_gumbel1 (const gsl_rng * R, double A, double B)
     This function returns  a random variate from the Type-1 Gumbel distribution.  The Type-1 Gumbel distribution function is,

          p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_gumbel1_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a Type-1 Gumbel distribution with parameters A and B, using the formula given above.

 -- Function: double gsl_cdf_gumbel1_P (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-1 Gumbel distribution with parameters A and B.

The Type-2 Gumbel Distribution
==============================

 -- Function: double gsl_ran_gumbel2 (const gsl_rng * R, double A, double B)
     This function returns a random variate from the Type-2 Gumbel distribution.  The Type-2 Gumbel distribution function is,

          p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

     for 0 < x < \infty.

 -- Function: double gsl_ran_gumbel2_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a Type-2 Gumbel distribution with parameters A and B, using the formula given above.

 -- Function: double gsl_cdf_gumbel2_P (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-2 Gumbel distribution with parameters A and B.

The Dirichlet Distribution
==========================

 -- Function: void gsl_ran_dirichlet (const gsl_rng * R, size_t K, const double ALPHA[], double THETA[])
     This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distribution function is

          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

     for $\theta_i \ge 0$ theta_i >= 0 and $\alpha_i \ge 0$ alpha_i >= 0.  The delta function ensures that \sum \theta_i = 1. The normalization factor Z is

          Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}

     The random variates are generated by sampling K values from gamma distributions with parameters $a=\alpha_i$, $b=1$ a=alpha_i, b=1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).

 -- Function: double gsl_ran_dirichlet_pdf (size_t K, const double ALPHA[], const double THETA[])
     This function computes the probability density $p(\theta_1, \ldots , \theta_K)$ p(\theta_1, ... , \theta_K) at THETA[K] for a Dirichlet distribution with parameters ALPHA[K], using the formula given above.

 -- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double ALPHA[], const double THETA[])
     This function computes the logarithm of the probability density $p(\theta_1, \ldots , \theta_K)$ p(\theta_1, ... , \theta_K) for a Dirichlet distribution with parameters ALPHA[K].

General Discrete Distributions
==============================

Given K discrete events with different probabilities P[k], produce a random value k consistent with its probability.

The obvious way to do this is to preprocess the probability list by generating a cumulative probability array with K+1 elements:

       C[0] = 0
     C[k+1] = C[k]+P[k].

Note that this construction produces C[K]=1.  Now choose a uniform deviate u between 0 and 1, and find the value of k such that $C[k] \le u < C[k+1]$ C[k] <= u < C[k+1]. Although this in principle requires of order \log K steps per random number generation, they are fast steps, and if you use something like \lfloor uK \rfloor as a starting point, you can often do pretty well.

But faster methods have been devised.  Again, the idea is to preprocess the probability list, and save the result in some form of lookup table; then the individual calls for a random discrete event can go rapidly. An approach invented by G. Marsaglia (Generating discrete random numbers in a computer, Comm ACM 6, 37--38 (1963)) is very clever, and readers interested in examples of good algorithm design are directed to this short and well-written paper.  Unfortunately, for large K, Marsaglia's lookup table can be quite large.

A much better approach is due to Alastair J. Walker (An efficient method for generating discrete random variables with general distributions, ACM Trans on Mathematical Software 3, 253--256 (1977); see also Knuth, v2, 3rd ed, p120--121,139).  This requires two lookup tables, one floating point and one integer, but both only of size K.  After preprocessing, the random numbers are generated in O(1) time, even for large K.  The preprocessing suggested by Walker requires O(K^2) effort, but that is not actually necessary, and the implementation provided here only takes O(K) effort.  In general, more preprocessing leads to faster generation of the individual random numbers, but a diminishing return is reached pretty early.  Knuth points out that the optimal preprocessing is combinatorially difficult for large K.

This method can be used to speed up some of the discrete random number generators below, such as the binomial distribution.  To use it for something like the Poisson Distribution, a modification would have to be made, since it only takes a finite set of K outcomes.

 -- Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K, const double * P)
     This function returns a pointer to a structure that contains the lookup table for the discrete random number generator.  The array P[] contains the probabilities of the discrete events; these array elements must all be positive, but they needn't add up to one (so you can think of them more generally as ``weights'')---the preprocessor will normalize appropriately. This return value is used as an argument for the gsl_ran_discrete function below.

 -- Function: size_t gsl_ran_discrete (const gsl_rng * R, const gsl_ran_discrete_t * G)
     After the preprocessor, above, has been called, you use this function to get the discrete random numbers.

 -- Function: double gsl_ran_discrete_pdf (size_t K, const gsl_ran_discrete_t * G)
     Returns the probability P[k] of observing the variable K. Since P[k] is not stored as part of the lookup table, it must be recomputed; this computation takes O(K), so if K is large and you care about the original array P[k] used to create the lookup table, then you should just keep this original array P[k] around.

 -- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * G)
     De-allocates the lookup table pointed to by G.

The Poisson Distribution
========================

 -- Function: unsigned int gsl_ran_poisson (const gsl_rng * R, double MU)
     This function returns a random integer from the Poisson distribution with mean MU.  The probability distribution for Poisson variates is,

          p(k) = {\mu^k \over k!} \exp(-\mu)

     for $k \ge 0$ k >= 0.

 -- Function: double gsl_ran_poisson_pdf (unsigned int K, double MU)
     This function computes the probability p(k) of obtaining  K from a Poisson distribution with mean MU, using the formula given above.

 -- Function: double gsl_cdf_poisson_P (unsigned int K, double MU)
 -- Function: double gsl_cdf_poisson_Q (unsigned int K, double MU)
     These functions compute the cumulative distribution functions P(k), Q(k) for the Poisson distribution with parameter MU.

The Bernoulli Distribution
==========================

 -- Function: unsigned int gsl_ran_bernoulli (const gsl_rng * R, double P)
     This function returns either 0 or 1, the result of a Bernoulli trial with probability P.  The probability distribution for a Bernoulli trial is,

          p(0) = 1 - p
          p(1) = p

 -- Function: double gsl_ran_bernoulli_pdf (unsigned int K, double P)
     This function computes the probability p(k) of obtaining K from a Bernoulli distribution with probability parameter P, using the formula given above.

The Binomial Distribution
=========================

 -- Function: unsigned int gsl_ran_binomial (const gsl_rng * R, double P, unsigned int N)
     This function returns a random integer from the binomial distribution, the number of successes in N independent trials with probability P.  The probability distribution for binomial variates is,

          p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}

     for $0 \le k \le n$ 0 <= k <= n.

 -- Function: double gsl_ran_binomial_pdf (unsigned int K, double P, unsigned int N)
     This function computes the probability p(k) of obtaining K from a binomial distribution with parameters P and N, using the formula given above.

 -- Function: double gsl_cdf_binomial_P (unsigned int K, double P, unsigned int N)
 -- Function: double gsl_cdf_binomial_Q (unsigned int K, double P, unsigned int N)
     These functions compute the cumulative distribution functions P(k), Q(k)  for the binomial distribution with parameters P and N.

The Multinomial Distribution
============================

 -- Function: void gsl_ran_multinomial (const gsl_rng * R, size_t K, unsigned int N, const double P[], unsigned int N[])

     This function computes a random sample N[] from the multinomial distribution formed by N trials from an underlying distribution P[K]. The distribution function for N[] is,

          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

     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 P[K] is not normalized then its entries will be treated as weights and normalized appropriately.  The arrays N[] and P[] must both be of length K.

     Random variates are generated using the conditional binomial method (see C.S. David, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205--217 for details).

 -- Function: double gsl_ran_multinomial_pdf (size_t K, const double P[], const unsigned int N[])
     This function computes the probability $P(n_1, n_2, \ldots, n_K)$ P(n_1, n_2, ..., n_K) of sampling N[K] from a multinomial distribution with parameters P[K], using the formula given above.

 -- Function: double gsl_ran_multinomial_lnpdf (size_t K, const double P[], const unsigned int N[])
     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 P[K].

The Negative Binomial Distribution
==================================

 -- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng * R, double P, double N)
     This function returns a random integer from the negative binomial distribution, the number of failures occurring before N successes in independent trials with probability P of success.  The probability distribution for negative binomial variates is,

          p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k

     Note that n is not required to be an integer.

 -- Function: double gsl_ran_negative_binomial_pdf (unsigned int K, double P, double N)
     This function computes the probability p(k) of obtaining K from a negative binomial distribution with parameters P and N, using the formula given above.

 -- Function: double gsl_cdf_negative_binomial_P (unsigned int K, double P, double N)
 -- Function: double gsl_cdf_negative_binomial_Q (unsigned int K, double P, double N)
     These functions compute the cumulative distribution functions P(k), Q(k) for the negative binomial distribution with parameters P and N.

The Pascal Distribution
=======================

 -- Function: unsigned int gsl_ran_pascal (const gsl_rng * R, double P, unsigned int N)
     This function returns a random integer from the Pascal distribution.  The Pascal distribution is simply a negative binomial distribution with an integer value of n.

          p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k

     for $k \ge 0$ k >= 0

 -- Function: double gsl_ran_pascal_pdf (unsigned int K, double P, unsigned int N)
     This function computes the probability p(k) of obtaining K from a Pascal distribution with parameters P and N, using the formula given above.

 -- Function: double gsl_cdf_pascal_P (unsigned int K, double P, unsigned int N)
 -- Function: double gsl_cdf_pascal_Q (unsigned int K, double P, unsigned int N)
     These functions compute the cumulative distribution functions P(k), Q(k) for the Pascal distribution with parameters P and N.

The Geometric Distribution
==========================

 -- Function: unsigned int gsl_ran_geometric (const gsl_rng * R, double P)
     This function returns a random integer from the geometric distribution, the number of independent trials with probability P until the first success.  The probability distribution for geometric variates is,

          p(k) =  p (1-p)^(k-1)

     for $k \ge 1$ k >= 1.  Note that the distribution begins with k=1 with this definition.  There is another convention in which the exponent k-1 is replaced by k.

 -- Function: double gsl_ran_geometric_pdf (unsigned int K, double P)
     This function computes the probability p(k) of obtaining K from a geometric distribution with probability parameter P, using the formula given above.

 -- Function: double gsl_cdf_geometric_P (unsigned int K, double P)
 -- Function: double gsl_cdf_geometric_Q (unsigned int K, double P)
     These functions compute the cumulative distribution functions P(k), Q(k) for the geometric distribution with parameter P.

The Hypergeometric Distribution
===============================

 -- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng * R, unsigned int N1, unsigned int N2, unsigned int T)
     This function returns a random integer from the hypergeometric distribution.  The probability distribution for hypergeometric random variates is,

          p(k) =  C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)

     where C(a,b) = a!/(b!(a-b)!) and $t \leq n_1 + n_2$ t <= n_1 + n_2.  The domain of k is $\hboxmax(0,t-n_2), \ldots, \hboxmin(t,n_1)$ max(0,t-n_2), ..., min(t,n_1).

     If a population contains n_1 elements of ``type 1'' and n_2 elements of ``type 2'' then the hypergeometric distribution gives the probability of obtaining k elements of ``type 1'' in t samples from the population without replacement.

 -- Function: double gsl_ran_hypergeometric_pdf (unsigned int K, unsigned int N1, unsigned int N2, unsigned int T)
     This function computes the probability p(k) of obtaining K from a hypergeometric distribution with parameters N1, N2, T, using the formula given above.

 -- Function: double gsl_cdf_hypergeometric_P (unsigned int K, unsigned int N1, unsigned int N2, unsigned int T)
 -- Function: double gsl_cdf_hypergeometric_Q (unsigned int K, unsigned int N1, unsigned int N2, unsigned int T)
     These functions compute the cumulative distribution functions P(k), Q(k) for the hypergeometric distribution with parameters N1, N2 and T.

The Logarithmic Distribution
============================

 -- Function: unsigned int gsl_ran_logarithmic (const gsl_rng * R, double P)
     This function returns a random integer from the logarithmic distribution.  The probability distribution for logarithmic random variates is,

          p(k) = {-1 \over \log(1-p)} {(p^k \over k)}

     for $k \ge 1$ k >= 1.

 -- Function: double gsl_ran_logarithmic_pdf (unsigned int K, double P)
     This function computes the probability p(k) of obtaining K from a logarithmic distribution with probability parameter P, using the formula given above.

Shuffling and Sampling
======================

The following functions allow the shuffling and sampling of a set of objects.  The algorithms rely on a random number generator as a source of randomness and a poor quality generator can lead to correlations in the output.  In particular it is important to avoid generators with a short period.  For more information see Knuth, v2, 3rd ed, Section 3.4.2, ``Random Sampling and Shuffling''.

 -- Function: void gsl_ran_shuffle (const gsl_rng * R, void * BASE, size_t N, size_t SIZE)

     This function randomly shuffles the order of N objects, each of size SIZE, stored in the array BASE[0..N-1].  The output of the random number generator R is used to produce the permutation.  The algorithm generates all possible n! permutations with equal probability, assuming a perfect source of random numbers.

     The following code shows how to shuffle the numbers from 0 to 51,

          int a[52];

          for (i = 0; i < 52; i++)
            {
              a[i] = i;
            }

          gsl_ran_shuffle (r, a, 52, sizeof (int));

 -- Function: int gsl_ran_choose (const gsl_rng * R, void * DEST, size_t K, void * SRC, size_t N, size_t SIZE)
     This function fills the array DEST[k] with K objects taken randomly from the N elements of the array SRC[0..N-1].  The objects are each of size SIZE.  The output of the random number generator R is used to make the selection.  The algorithm ensures all possible samples are equally likely, assuming a perfect source of randomness.

     The objects are sampled without replacement, thus each object can only appear once in DEST[k].  It is required that K be less than or equal to n.  The objects in DEST will be in the same relative order as those in SRC.  You will need to call gsl_ran_shuffle(r, dest, n, size) if you want to randomize the order.

     The following code shows how to select a random sample of three unique numbers from the set 0 to 99,

          double a[3], b[100];

          for (i = 0; i < 100; i++)
            {
              b[i] = (double) i;
            }

          gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

 -- Function: void gsl_ran_sample (const gsl_rng * R, void * DEST, size_t K, void * SRC, size_t N, size_t SIZE)
     This function is like gsl_ran_choose but samples K items from the original array of N items SRC with replacement, so the same object can appear more than once in the output sequence DEST.  There is no requirement that K be less than N in this case.

Examples
========

The following program demonstrates the use of a random number generator to produce variates from a distribution.  It prints 10 samples from the Poisson distribution with a mean of 3.

          [file randpoisson.c not included]

If the library and header files are installed under /usr/local (the default location) then the program can be compiled with these options,

     $ gcc -Wall demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

     $ ./a.out
          [file randpoisson.out not included]

The variates depend on the seed used by the generator.  The seed for the default generator type gsl_rng_default can be changed with the GSL_RNG_SEED environment variable to produce a different stream of variates,

     $ GSL_RNG_SEED=123 ./a.out
          [file randpoisson.2.out not included]

The following program generates a random walk in two dimensions.

          [file randwalk.c not included]

Here is the output from the program, three 10-step random walks from the origin,

The following program computes the upper and lower cumulative distribution functions for the standard normal distribution at x=2.

          [file cdf.c not included]

Here is the output of the program,

          [file cdf.out not included]

References and Further Reading
==============================

For an encyclopaedic coverage of the subject readers are advised to consult the book Non-Uniform Random Variate Generation by Luc Devroye.  It covers every imaginable distribution and provides hundreds of algorithms.

  Luc Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, ISBN 0-387-96305-7.

The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.

  Donald E. Knuth, The Art of Computer Programming: Seminumerical Algorithms (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

The Particle Data Group provides a short review of techniques for generating distributions of random numbers in the ``Monte Carlo'' section of its Annual Review of Particle Physics.

  Review of Particle Properties R.M. Barnett et al., Physical Review D54, 1 (1996) http://pdg.lbl.gov/.

The Review of Particle Physics is available online in postscript and pdf format.

An overview of methods used to compute cumulative distribution functions can be found in Statistical Computing by W.J. Kennedy and J.E. Gentle. Another general reference is Elements of Statistical Computing by R.A. Thisted.

  William E. Kennedy and James E. Gentle, Statistical Computing (1980), Marcel Dekker, ISBN 0-8247-6898-1.

  Ronald A. Thisted, Elements of Statistical Computing (1988), Chapman & Hall, ISBN 0-412-01371-1.

The cumulative distribution functions for the Gaussian distribution are based on the following papers,

  Rational Chebyshev Approximations Using Linear Equations, W.J. Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242--251 (1968).

  Rational Chebyshev Approximations for the Error Function, W.J. Cody. Mathematics of Computation 23, n107, 631--637 (July 1969).