#[1]Index [2]Search [3]Statistics [4]Quasi-Random Sequences

   [5]GSL
   2.8
   ____________________
     * [6]Introduction
     * [7]Using the Library
     * [8]Error Handling
     * [9]Mathematical Functions
     * [10]Complex Numbers
     * [11]Polynomials
     * [12]Special Functions
     * [13]Vectors and Matrices
     * [14]Permutations
     * [15]Combinations
     * [16]Multisets
     * [17]Sorting
     * [18]BLAS Support
     * [19]Linear Algebra
     * [20]Eigensystems
     * [21]Fast Fourier Transforms (FFTs)
     * [22]Numerical Integration
     * [23]Random Number Generation
     * [24]Quasi-Random Sequences
     * [25]Random Number Distributions
          + [26]Introduction
          + [27]The Gaussian Distribution
          + [28]The Gaussian Tail Distribution
          + [29]The Bivariate Gaussian Distribution
          + [30]The Multivariate Gaussian Distribution
          + [31]The Exponential Distribution
          + [32]The Laplace Distribution
          + [33]The Exponential Power Distribution
          + [34]The Cauchy Distribution
          + [35]The Rayleigh Distribution
          + [36]The Rayleigh Tail Distribution
          + [37]The Landau Distribution
          + [38]The Levy alpha-Stable Distributions
          + [39]The Levy skew alpha-Stable Distribution
          + [40]The Gamma Distribution
          + [41]The Flat (Uniform) Distribution
          + [42]The Lognormal Distribution
          + [43]The Chi-squared Distribution
          + [44]The F-distribution
          + [45]The t-distribution
          + [46]The Beta Distribution
          + [47]The Logistic Distribution
          + [48]The Pareto Distribution
          + [49]Spherical Vector Distributions
          + [50]The Weibull Distribution
          + [51]The Type-1 Gumbel Distribution
          + [52]The Type-2 Gumbel Distribution
          + [53]The Dirichlet Distribution
          + [54]General Discrete Distributions
          + [55]The Poisson Distribution
          + [56]The Bernoulli Distribution
          + [57]The Binomial Distribution
          + [58]The Multinomial Distribution
          + [59]The Negative Binomial Distribution
          + [60]The Pascal Distribution
          + [61]The Geometric Distribution
          + [62]The Hypergeometric Distribution
          + [63]The Logarithmic Distribution
          + [64]The Wishart Distribution
          + [65]Shuffling and Sampling
          + [66]Examples
          + [67]References and Further Reading
     * [68]Statistics
     * [69]Running Statistics
     * [70]Moving Window Statistics
     * [71]Digital Filtering
     * [72]Histograms
     * [73]N-tuples
     * [74]Monte Carlo Integration
     * [75]Simulated Annealing
     * [76]Ordinary Differential Equations
     * [77]Interpolation
     * [78]Numerical Differentiation
     * [79]Chebyshev Approximations
     * [80]Series Acceleration
     * [81]Wavelet Transforms
     * [82]Discrete Hankel Transforms
     * [83]One Dimensional Root-Finding
     * [84]One Dimensional Minimization
     * [85]Multidimensional Root-Finding
     * [86]Multidimensional Minimization
     * [87]Linear Least-Squares Fitting
     * [88]Nonlinear Least-Squares Fitting
     * [89]Basis Splines
     * [90]Sparse Matrices
     * [91]Sparse BLAS Support
     * [92]Sparse Linear Algebra
     * [93]Physical Constants
     * [94]IEEE floating-point arithmetic
     * [95]Debugging Numerical Programs
     * [96]Contributors to GSL
     * [97]Autoconf Macros
     * [98]GSL CBLAS Library
     * [99]GNU General Public License
     * [100]GNU Free Documentation License

   [101]GSL
     * »
     * Random Number Distributions
     * [102]View page source

   [103]Next [104]Previous
     ____________________________________________________________________________

Random Number Distributions[105]¶

   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 \times 10^9

   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[106]¶

   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(x) 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 \le Q(x) \le 1 .

   The inverse cumulative distributions, x = P^{-1}(P) and 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 \le 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[107]¶

   double gsl_ran_gaussian(const [108]gsl_rng *r, double sigma)[109]¶
          This function returns a Gaussian random variate, with mean zero and
          standard deviation [110]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 [111]gsl_ran_gaussian() to obtain a Gaussian
          distribution with mean \mu . This function uses the Box-Muller algorithm
          which requires two calls to the random number generator [112]r.

   double gsl_ran_gaussian_pdf(double x, double sigma)[113]¶
          This function computes the probability density p(x) at [114]x for a
          Gaussian distribution with standard deviation [115]sigma, using the
          formula given above.

          _images/rand-gaussian.png

   double gsl_ran_gaussian_ziggurat(const [116]gsl_rng *r, double sigma)[117]¶

   double gsl_ran_gaussian_ratio_method(const [118]gsl_rng *r, double sigma)[119]¶
          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.

   double gsl_ran_ugaussian(const [120]gsl_rng *r)[121]¶

   double gsl_ran_ugaussian_pdf(double x)[122]¶

   double gsl_ran_ugaussian_ratio_method(const [123]gsl_rng *r)[124]¶
          These functions compute results for the unit Gaussian distribution. They
          are equivalent to the functions above with a standard deviation of one,
          sigma = 1.

   double gsl_cdf_gaussian_P(double x, double sigma)[125]¶

   double gsl_cdf_gaussian_Q(double x, double sigma)[126]¶

   double gsl_cdf_gaussian_Pinv(double P, double sigma)[127]¶

   double gsl_cdf_gaussian_Qinv(double Q, double sigma)[128]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Gaussian distribution with standard deviation
          [129]sigma.

   double gsl_cdf_ugaussian_P(double x)[130]¶

   double gsl_cdf_ugaussian_Q(double x)[131]¶

   double gsl_cdf_ugaussian_Pinv(double P)[132]¶

   double gsl_cdf_ugaussian_Qinv(double Q)[133]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the unit Gaussian distribution.

The Gaussian Tail Distribution[134]¶

   double gsl_ran_gaussian_tail(const [135]gsl_rng *r, double a, double sigma)[136]¶
          This function provides random variates from the upper tail of a Gaussian
          distribution with standard deviation [137]sigma. The values returned are
          larger than the lower limit [138]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 \over 2} \hbox{erfc}\left({a \over \sqrt{2
          \sigma^2}}\right).

   double gsl_ran_gaussian_tail_pdf(double x, double a, double sigma)[139]¶
          This function computes the probability density p(x) at [140]x for a
          Gaussian tail distribution with standard deviation [141]sigma and lower
          limit [142]a, using the formula given above.

          _images/rand-gaussian-tail.png

   double gsl_ran_ugaussian_tail(const [143]gsl_rng *r, double a)[144]¶

   double gsl_ran_ugaussian_tail_pdf(double x, double a)[145]¶
          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[146]¶

   void gsl_ran_bivariate_gaussian(const [147]gsl_rng *r, double sigma_x, double
          sigma_y, double rho, double *x, double *y)[148]¶
          This function generates a pair of correlated Gaussian variates, with mean
          zero, correlation coefficient [149]rho and standard deviations
          [150]sigma_x and [151]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
          \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

          for x,y in the range -\infty to +\infty . The correlation coefficient
          [152]rho should lie between 1 and -1 .

   double gsl_ran_bivariate_gaussian_pdf(double x, double y, double sigma_x, double
          sigma_y, double rho)[153]¶
          This function computes the probability density p(x,y) at ([154]x, [155]y)
          for a bivariate Gaussian distribution with standard deviations
          [156]sigma_x, [157]sigma_y and correlation coefficient [158]rho, using the
          formula given above.

          _images/rand-bivariate-gaussian.png

The Multivariate Gaussian Distribution[159]¶

   int gsl_ran_multivariate_gaussian(const [160]gsl_rng *r, const [161]gsl_vector
          *mu, const [162]gsl_matrix *L, [163]gsl_vector *result)[164]¶
          This function generates a random vector satisfying the k -dimensional
          multivariate Gaussian distribution with mean \mu and variance-covariance
          matrix \Sigma . On input, the k -vector \mu is given in [165]mu, and the
          Cholesky factor of the k -by- k matrix \Sigma = L L^T is given in the
          lower triangle of [166]L, as output from
          [167]gsl_linalg_cholesky_decomp(). The random vector is stored in
          [168]result on output. The probability distribution for multivariate
          Gaussian random variates is

          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

   int gsl_ran_multivariate_gaussian_pdf(const [169]gsl_vector *x, const
          [170]gsl_vector *mu, const [171]gsl_matrix *L, double *result,
          [172]gsl_vector *work)[173]¶

   int gsl_ran_multivariate_gaussian_log_pdf(const [174]gsl_vector *x, const
          [175]gsl_vector *mu, const [176]gsl_matrix *L, double *result,
          [177]gsl_vector *work)[178]¶
          These functions compute p(x) or \log{p(x)} at the point [179]x, using mean
          vector [180]mu and variance-covariance matrix specified by its Cholesky
          factor [181]L using the formula above. Additional workspace of length k is
          required in [182]work.

   int gsl_ran_multivariate_gaussian_mean(const [183]gsl_matrix *X, [184]gsl_vector
          *mu_hat)[185]¶
          Given a set of n samples X_j from a k -dimensional multivariate Gaussian
          distribution, this function computes the maximum likelihood estimate of
          the mean of the distribution, given by

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

          The samples X_1,X_2,\dots,X_n are given in the n -by- k matrix [186]X, and
          the maximum likelihood estimate of the mean is stored in [187]mu_hat on
          output.

   int gsl_ran_multivariate_gaussian_vcov(const [188]gsl_matrix *X, [189]gsl_matrix
          *sigma_hat)[190]¶
          Given a set of n samples X_j from a k -dimensional multivariate Gaussian
          distribution, this function computes the maximum likelihood estimate of
          the variance-covariance matrix of the distribution, given by

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

          The samples X_1,X_2,\dots,X_n are given in the n -by- k matrix [191]X and
          the maximum likelihood estimate of the variance-covariance matrix is
          stored in [192]sigma_hat on output.

The Exponential Distribution[193]¶

   double gsl_ran_exponential(const [194]gsl_rng *r, double mu)[195]¶
          This function returns a random variate from the exponential distribution
          with mean [196]mu. The distribution is,

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

          for x \ge 0 .

   double gsl_ran_exponential_pdf(double x, double mu)[197]¶
          This function computes the probability density p(x) at [198]x for an
          exponential distribution with mean [199]mu, using the formula given above.

          _images/rand-exponential.png

   double gsl_cdf_exponential_P(double x, double mu)[200]¶

   double gsl_cdf_exponential_Q(double x, double mu)[201]¶

   double gsl_cdf_exponential_Pinv(double P, double mu)[202]¶

   double gsl_cdf_exponential_Qinv(double Q, double mu)[203]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the exponential distribution with mean [204]mu.

The Laplace Distribution[205]¶

   double gsl_ran_laplace(const [206]gsl_rng *r, double a)[207]¶
          This function returns a random variate from the Laplace distribution with
          width [208]a. The distribution is,

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

          for -\infty < x < \infty .

   double gsl_ran_laplace_pdf(double x, double a)[209]¶
          This function computes the probability density p(x) at [210]x for a
          Laplace distribution with width [211]a, using the formula given above.

          _images/rand-laplace.png

   double gsl_cdf_laplace_P(double x, double a)[212]¶

   double gsl_cdf_laplace_Q(double x, double a)[213]¶

   double gsl_cdf_laplace_Pinv(double P, double a)[214]¶

   double gsl_cdf_laplace_Qinv(double Q, double a)[215]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Laplace distribution with width [216]a.

The Exponential Power Distribution[217]¶

   double gsl_ran_exppow(const [218]gsl_rng *r, double a, double b)[219]¶
          This function returns a random variate from the exponential power
          distribution with scale parameter [220]a and exponent [221]b. The
          distribution is,

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

          for x \ge 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 = \sqrt{2}
          \sigma .

   double gsl_ran_exppow_pdf(double x, double a, double b)[222]¶
          This function computes the probability density p(x) at [223]x for an
          exponential power distribution with scale parameter [224]a and exponent
          [225]b, using the formula given above.

          _images/rand-exppow.png

   double gsl_cdf_exppow_P(double x, double a, double b)[226]¶

   double gsl_cdf_exppow_Q(double x, double a, double b)[227]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          for the exponential power distribution with parameters [228]a and [229]b.

The Cauchy Distribution[230]¶

   double gsl_ran_cauchy(const [231]gsl_rng *r, double a)[232]¶
          This function returns a random variate from the Cauchy distribution with
          scale parameter [233]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.

   double gsl_ran_cauchy_pdf(double x, double a)[234]¶
          This function computes the probability density p(x) at [235]x for a Cauchy
          distribution with scale parameter [236]a, using the formula given above.

          _images/rand-cauchy.png

   double gsl_cdf_cauchy_P(double x, double a)[237]¶

   double gsl_cdf_cauchy_Q(double x, double a)[238]¶

   double gsl_cdf_cauchy_Pinv(double P, double a)[239]¶

   double gsl_cdf_cauchy_Qinv(double Q, double a)[240]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Cauchy distribution with scale parameter
          [241]a.

The Rayleigh Distribution[242]¶

   double gsl_ran_rayleigh(const [243]gsl_rng *r, double sigma)[244]¶
          This function returns a random variate from the Rayleigh distribution with
          scale parameter [245]sigma. The distribution is,

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

          for x > 0 .

   double gsl_ran_rayleigh_pdf(double x, double sigma)[246]¶
          This function computes the probability density p(x) at [247]x for a
          Rayleigh distribution with scale parameter [248]sigma, using the formula
          given above.

          _images/rand-rayleigh.png

   double gsl_cdf_rayleigh_P(double x, double sigma)[249]¶

   double gsl_cdf_rayleigh_Q(double x, double sigma)[250]¶

   double gsl_cdf_rayleigh_Pinv(double P, double sigma)[251]¶

   double gsl_cdf_rayleigh_Qinv(double Q, double sigma)[252]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Rayleigh distribution with scale parameter
          [253]sigma.

The Rayleigh Tail Distribution[254]¶

   double gsl_ran_rayleigh_tail(const [255]gsl_rng *r, double a, double sigma)[256]¶
          This function returns a random variate from the tail of the Rayleigh
          distribution with scale parameter [257]sigma and a lower limit of [258]a.
          The distribution is,

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

          for x > a .

   double gsl_ran_rayleigh_tail_pdf(double x, double a, double sigma)[259]¶
          This function computes the probability density p(x) at [260]x for a
          Rayleigh tail distribution with scale parameter [261]sigma and lower limit
          [262]a, using the formula given above.

          _images/rand-rayleigh-tail.png

The Landau Distribution[263]¶

   double gsl_ran_landau(const [264]gsl_rng *r)[265]¶
          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 \over {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).

   double gsl_ran_landau_pdf(double x)[266]¶
          This function computes the probability density p(x) at [267]x for the
          Landau distribution using an approximation to the formula given above.

          _images/rand-landau.png

The Levy alpha-Stable Distributions[268]¶

   double gsl_ran_levy(const [269]gsl_rng *r, double c, double alpha)[270]¶
          This function returns a random variate from the Levy symmetric stable
          distribution with scale [271]c and exponent [272]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 = \sqrt{2} c . For \alpha < 1 the tails of the
          distribution become extremely wide.

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

          _images/rand-levy.png

The Levy skew alpha-Stable Distribution[273]¶

   double gsl_ran_levy_skew(const [274]gsl_rng *r, double c, double alpha, double
          beta)[275]¶
          This function returns a random variate from the Levy skew stable
          distribution with scale [276]c, exponent [277]alpha and skewness parameter
          [278]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 \sgn(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 = \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 .

   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) .
   _images/rand-levyskew.png

The Gamma Distribution[279]¶

   double gsl_ran_gamma(const [280]gsl_rng *r, double a, double b)[281]¶
          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 [282]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.

   double gsl_ran_gamma_knuth(const [283]gsl_rng *r, double a, double b)[284]¶
          This function returns a gamma variate using the algorithms from Knuth (vol
          2).

   double gsl_ran_gamma_pdf(double x, double a, double b)[285]¶
          This function computes the probability density p(x) at [286]x for a gamma
          distribution with parameters [287]a and [288]b, using the formula given
          above.

          _images/rand-gamma.png

   double gsl_cdf_gamma_P(double x, double a, double b)[289]¶

   double gsl_cdf_gamma_Q(double x, double a, double b)[290]¶

   double gsl_cdf_gamma_Pinv(double P, double a, double b)[291]¶

   double gsl_cdf_gamma_Qinv(double Q, double a, double b)[292]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the gamma distribution with parameters [293]a and
          [294]b.

The Flat (Uniform) Distribution[295]¶

   double gsl_ran_flat(const [296]gsl_rng *r, double a, double b)[297]¶
          This function returns a random variate from the flat (uniform)
          distribution from [298]a to [299]b. The distribution is,

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

          if a \le x < b and 0 otherwise.

   double gsl_ran_flat_pdf(double x, double a, double b)[300]¶
          This function computes the probability density p(x) at [301]x for a
          uniform distribution from [302]a to [303]b, using the formula given above.

          _images/rand-flat.png

   double gsl_cdf_flat_P(double x, double a, double b)[304]¶

   double gsl_cdf_flat_Q(double x, double a, double b)[305]¶

   double gsl_cdf_flat_Pinv(double P, double a, double b)[306]¶

   double gsl_cdf_flat_Qinv(double Q, double a, double b)[307]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for a uniform distribution from [308]a to [309]b.

The Lognormal Distribution[310]¶

   double gsl_ran_lognormal(const [311]gsl_rng *r, double zeta, double sigma)[312]¶
          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 .

   double gsl_ran_lognormal_pdf(double x, double zeta, double sigma)[313]¶
          This function computes the probability density p(x) at [314]x for a
          lognormal distribution with parameters [315]zeta and [316]sigma, using the
          formula given above.

          _images/rand-lognormal.png

   double gsl_cdf_lognormal_P(double x, double zeta, double sigma)[317]¶

   double gsl_cdf_lognormal_Q(double x, double zeta, double sigma)[318]¶

   double gsl_cdf_lognormal_Pinv(double P, double zeta, double sigma)[319]¶

   double gsl_cdf_lognormal_Qinv(double Q, double zeta, double sigma)[320]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the lognormal distribution with parameters
          [321]zeta and [322]sigma.

The Chi-squared Distribution[323]¶

   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.

   double gsl_ran_chisq(const [324]gsl_rng *r, double nu)[325]¶
          This function returns a random variate from the chi-squared distribution
          with [326]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 .

   double gsl_ran_chisq_pdf(double x, double nu)[327]¶
          This function computes the probability density p(x) at [328]x for a
          chi-squared distribution with [329]nu degrees of freedom, using the
          formula given above.

          _images/rand-chisq.png

   double gsl_cdf_chisq_P(double x, double nu)[330]¶

   double gsl_cdf_chisq_Q(double x, double nu)[331]¶

   double gsl_cdf_chisq_Pinv(double P, double nu)[332]¶

   double gsl_cdf_chisq_Qinv(double Q, double nu)[333]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the chi-squared distribution with [334]nu degrees
          of freedom.

The F-distribution[335]¶

   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) .

   double gsl_ran_fdist(const [336]gsl_rng *r, double nu1, double nu2)[337]¶
          This function returns a random variate from the F-distribution with
          degrees of freedom [338]nu1 and [339]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 .

   double gsl_ran_fdist_pdf(double x, double nu1, double nu2)[340]¶
          This function computes the probability density p(x) at [341]x for an
          F-distribution with [342]nu1 and [343]nu2 degrees of freedom, using the
          formula given above.

          _images/rand-fdist.png

   double gsl_cdf_fdist_P(double x, double nu1, double nu2)[344]¶

   double gsl_cdf_fdist_Q(double x, double nu1, double nu2)[345]¶

   double gsl_cdf_fdist_Pinv(double P, double nu1, double nu2)[346]¶

   double gsl_cdf_fdist_Qinv(double Q, double nu1, double nu2)[347]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the F-distribution with [348]nu1 and [349]nu2
          degrees of freedom.

The t-distribution[350]¶

   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.

   double gsl_ran_tdist(const [351]gsl_rng *r, double nu)[352]¶
          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 .

   double gsl_ran_tdist_pdf(double x, double nu)[353]¶
          This function computes the probability density p(x) at [354]x for a
          t-distribution with [355]nu degrees of freedom, using the formula given
          above.

          _images/rand-tdist.png

   double gsl_cdf_tdist_P(double x, double nu)[356]¶

   double gsl_cdf_tdist_Q(double x, double nu)[357]¶

   double gsl_cdf_tdist_Pinv(double P, double nu)[358]¶

   double gsl_cdf_tdist_Qinv(double Q, double nu)[359]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the t-distribution with [360]nu degrees of freedom.

The Beta Distribution[361]¶

   double gsl_ran_beta(const [362]gsl_rng *r, double a, double b)[363]¶
          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 .

   double gsl_ran_beta_pdf(double x, double a, double b)[364]¶
          This function computes the probability density p(x) at [365]x for a beta
          distribution with parameters [366]a and [367]b, using the formula given
          above.

          _images/rand-beta.png

   double gsl_cdf_beta_P(double x, double a, double b)[368]¶

   double gsl_cdf_beta_Q(double x, double a, double b)[369]¶

   double gsl_cdf_beta_Pinv(double P, double a, double b)[370]¶

   double gsl_cdf_beta_Qinv(double Q, double a, double b)[371]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the beta distribution with parameters [372]a and
          [373]b.

The Logistic Distribution[374]¶

   double gsl_ran_logistic(const [375]gsl_rng *r, double a)[376]¶
          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 .

   double gsl_ran_logistic_pdf(double x, double a)[377]¶
          This function computes the probability density p(x) at [378]x for a
          logistic distribution with scale parameter [379]a, using the formula given
          above.

          _images/rand-logistic.png

   double gsl_cdf_logistic_P(double x, double a)[380]¶

   double gsl_cdf_logistic_Q(double x, double a)[381]¶

   double gsl_cdf_logistic_Pinv(double P, double a)[382]¶

   double gsl_cdf_logistic_Qinv(double Q, double a)[383]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the logistic distribution with scale parameter
          [384]a.

The Pareto Distribution[385]¶

   double gsl_ran_pareto(const [386]gsl_rng *r, double a, double b)[387]¶
          This function returns a random variate from the Pareto distribution of
          order [388]a. The distribution function is,

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

          for x \ge b .

   double gsl_ran_pareto_pdf(double x, double a, double b)[389]¶
          This function computes the probability density p(x) at [390]x for a Pareto
          distribution with exponent [391]a and scale [392]b, using the formula
          given above.

          _images/rand-pareto.png

   double gsl_cdf_pareto_P(double x, double a, double b)[393]¶

   double gsl_cdf_pareto_Q(double x, double a, double b)[394]¶

   double gsl_cdf_pareto_Pinv(double P, double a, double b)[395]¶

   double gsl_cdf_pareto_Qinv(double Q, double a, double b)[396]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Pareto distribution with exponent [397]a and
          scale [398]b.

Spherical Vector Distributions[399]¶

   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.

   void gsl_ran_dir_2d(const [400]gsl_rng *r, double *x, double *y)[401]¶

   void gsl_ran_dir_2d_trig_method(const [402]gsl_rng *r, double *x, double
          *y)[403]¶
          This function returns a random direction vector v = ([404]x, [405]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 [406]x and [407]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 [408]x and
          [409]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 \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) .

   void gsl_ran_dir_3d(const [410]gsl_rng *r, double *x, double *y, double *z)[411]¶
          This function returns a random direction vector v = ([412]x, [413]y,
          [414]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).

   void gsl_ran_dir_nd(const [415]gsl_rng *r, size_t n, double *x)[416]¶
          This function returns a random direction vector v = (x_1,x_2,\ldots,x_n)
          in [417]n dimensions. The vector is normalized such that |v|^2 = x_1^2 +
          x_2^2 + \cdots + 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[418]¶

   double gsl_ran_weibull(const [419]gsl_rng *r, double a, double b)[420]¶
          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 .

   double gsl_ran_weibull_pdf(double x, double a, double b)[421]¶
          This function computes the probability density p(x) at [422]x for a
          Weibull distribution with scale [423]a and exponent [424]b, using the
          formula given above.

          _images/rand-weibull.png

   double gsl_cdf_weibull_P(double x, double a, double b)[425]¶

   double gsl_cdf_weibull_Q(double x, double a, double b)[426]¶

   double gsl_cdf_weibull_Pinv(double P, double a, double b)[427]¶

   double gsl_cdf_weibull_Qinv(double Q, double a, double b)[428]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Weibull distribution with scale [429]a and
          exponent [430]b.

The Type-1 Gumbel Distribution[431]¶

   double gsl_ran_gumbel1(const [432]gsl_rng *r, double a, double b)[433]¶
          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 .

   double gsl_ran_gumbel1_pdf(double x, double a, double b)[434]¶
          This function computes the probability density p(x) at [435]x for a Type-1
          Gumbel distribution with parameters [436]a and [437]b, using the formula
          given above.

          _images/rand-gumbel1.png

   double gsl_cdf_gumbel1_P(double x, double a, double b)[438]¶

   double gsl_cdf_gumbel1_Q(double x, double a, double b)[439]¶

   double gsl_cdf_gumbel1_Pinv(double P, double a, double b)[440]¶

   double gsl_cdf_gumbel1_Qinv(double Q, double a, double b)[441]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Type-1 Gumbel distribution with parameters
          [442]a and [443]b.

The Type-2 Gumbel Distribution[444]¶

   double gsl_ran_gumbel2(const [445]gsl_rng *r, double a, double b)[446]¶
          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 .

   double gsl_ran_gumbel2_pdf(double x, double a, double b)[447]¶
          This function computes the probability density p(x) at [448]x for a Type-2
          Gumbel distribution with parameters [449]a and [450]b, using the formula
          given above.

          _images/rand-gumbel2.png

   double gsl_cdf_gumbel2_P(double x, double a, double b)[451]¶

   double gsl_cdf_gumbel2_Q(double x, double a, double b)[452]¶

   double gsl_cdf_gumbel2_Pinv(double P, double a, double b)[453]¶

   double gsl_cdf_gumbel2_Qinv(double Q, double a, double b)[454]¶
          These functions compute the cumulative distribution functions P(x) , Q(x)
          and their inverses for the Type-2 Gumbel distribution with parameters
          [455]a and [456]b.

The Dirichlet Distribution[457]¶

   void gsl_ran_dirichlet(const [458]gsl_rng *r, size_t K, const double alpha[],
          double theta[])[459]¶
          This function returns an array of [460]K random variates from a Dirichlet
          distribution of order [461]K-1. The distribution function is

          p(\theta_1,\ldots,\theta_K) \, d\theta_1 \cdots d\theta_K = {1 \over Z}
          \prod_{i=1}^{K} \theta_i^{\alpha_i - 1} \; \delta(1 -\sum_{i=1}^K
          \theta_i) d\theta_1 \cdots d\theta_K

          for \theta_i \ge 0 and \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) \over \Gamma( \sum_{i=1}^K \alpha_i)}

          The random variates are generated by sampling [462]K values from gamma
          distributions with parameters a=\alpha_i$, $b=1 , and renormalizing. See
          A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).

   double gsl_ran_dirichlet_pdf(size_t K, const double alpha[], const double
          theta[])[463]¶
          This function computes the probability density p(\theta_1, \ldots ,
          \theta_K) at theta[K] for a Dirichlet distribution with parameters
          alpha[K], using the formula given above.

   double gsl_ran_dirichlet_lnpdf(size_t K, const double alpha[], const double
          theta[])[464]¶
          This function computes the logarithm of the probability density
          p(\theta_1, \ldots , \theta_K) for a Dirichlet distribution with
          parameters alpha[K].

General Discrete Distributions[465]¶

   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] . 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 variables 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.

   type gsl_ran_discrete_t[466]¶
          This structure contains the lookup table for the discrete random number
          generator.

   [467]gsl_ran_discrete_t *gsl_ran_discrete_preproc(size_t K, const double
          *P)[468]¶
          This function returns a pointer to a structure that contains the lookup
          table for the discrete random number generator. The array [469]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 [470]gsl_ran_discrete()
          function below.

   size_t gsl_ran_discrete(const [471]gsl_rng *r, const [472]gsl_ran_discrete_t
          *g)[473]¶
          After the preprocessor, above, has been called, you use this function to
          get the discrete random numbers.

   double gsl_ran_discrete_pdf(size_t k, const [474]gsl_ran_discrete_t *g)[475]¶
          Returns the probability P[k] of observing the variable [476]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.

   void gsl_ran_discrete_free([477]gsl_ran_discrete_t *g)[478]¶
          De-allocates the lookup table pointed to by [479]g.

The Poisson Distribution[480]¶

   unsigned int gsl_ran_poisson(const [481]gsl_rng *r, double mu)[482]¶
          This function returns a random integer from the Poisson distribution with
          mean [483]mu. The probability distribution for Poisson variates is,

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

          for k \ge 0 .

   double gsl_ran_poisson_pdf(unsigned int k, double mu)[484]¶
          This function computes the probability p(k) of obtaining [485]k from a
          Poisson distribution with mean [486]mu, using the formula given above.

          _images/rand-poisson.png

   double gsl_cdf_poisson_P(unsigned int k, double mu)[487]¶

   double gsl_cdf_poisson_Q(unsigned int k, double mu)[488]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the Poisson distribution with parameter [489]mu.

The Bernoulli Distribution[490]¶

   unsigned int gsl_ran_bernoulli(const [491]gsl_rng *r, double p)[492]¶
          This function returns either 0 or 1, the result of a Bernoulli trial with
          probability [493]p. The probability distribution for a Bernoulli trial is,

          p(0) & = 1 - p \\ p(1) & = p

   double gsl_ran_bernoulli_pdf(unsigned int k, double p)[494]¶
          This function computes the probability p(k) of obtaining [495]k from a
          Bernoulli distribution with probability parameter [496]p, using the
          formula given above.

          _images/rand-bernoulli.png

The Binomial Distribution[497]¶

   unsigned int gsl_ran_binomial(const [498]gsl_rng *r, double p, unsigned int
          n)[499]¶
          This function returns a random integer from the binomial distribution, the
          number of successes in [500]n independent trials with probability [501]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 .

   double gsl_ran_binomial_pdf(unsigned int k, double p, unsigned int n)[502]¶
          This function computes the probability p(k) of obtaining [503]k from a
          binomial distribution with parameters [504]p and [505]n, using the formula
          given above.

          _images/rand-binomial.png

   double gsl_cdf_binomial_P(unsigned int k, double p, unsigned int n)[506]¶

   double gsl_cdf_binomial_Q(unsigned int k, double p, unsigned int n)[507]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the binomial distribution with parameters [508]p and [509]n.

The Multinomial Distribution[510]¶

   void gsl_ran_multinomial(const [511]gsl_rng *r, size_t K, unsigned int N, const
          double p[], unsigned int n[])[512]¶
          This function computes a random sample [513]n from the multinomial
          distribution formed by [514]N trials from an underlying distribution p[K].
          The distribution function for [515]n is,

          P(n_1, n_2,\cdots, n_K) = {{ N!}\over{n_1 ! n_2 ! \cdots n_K !}} \,
          p_1^{n_1} p_2^{n_2} \cdots p_K^{n_K}

          where (n_1, n_2, \ldots, n_K) are nonnegative integers with \sum_{k=1}^{K}
          n_k = N , and (p_1, p_2, \ldots, 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 [516]n and
          [517]p must both be of length [518]K.

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

   double gsl_ran_multinomial_pdf(size_t K, const double p[], const unsigned int
          n[])[519]¶
          This function computes the probability P(n_1, n_2, \ldots, n_K) of
          sampling n[K] from a multinomial distribution with parameters p[K], using
          the formula given above.

   double gsl_ran_multinomial_lnpdf(size_t K, const double p[], const unsigned int
          n[])[520]¶
          This function returns the logarithm of the probability for the multinomial
          distribution P(n_1, n_2, \ldots, n_K) with parameters p[K].

The Negative Binomial Distribution[521]¶

   unsigned int gsl_ran_negative_binomial(const [522]gsl_rng *r, double p, double
          n)[523]¶
          This function returns a random integer from the negative binomial
          distribution, the number of failures occurring before [524]n successes in
          independent trials with probability [525]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.

   double gsl_ran_negative_binomial_pdf(unsigned int k, double p, double n)[526]¶
          This function computes the probability p(k) of obtaining [527]k from a
          negative binomial distribution with parameters [528]p and [529]n, using
          the formula given above.

          _images/rand-nbinomial.png

   double gsl_cdf_negative_binomial_P(unsigned int k, double p, double n)[530]¶

   double gsl_cdf_negative_binomial_Q(unsigned int k, double p, double n)[531]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the negative binomial distribution with parameters [532]p and [533]n.

The Pascal Distribution[534]¶

   unsigned int gsl_ran_pascal(const [535]gsl_rng *r, double p, unsigned int
          n)[536]¶
          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 .

   double gsl_ran_pascal_pdf(unsigned int k, double p, unsigned int n)[537]¶
          This function computes the probability p(k) of obtaining [538]k from a
          Pascal distribution with parameters [539]p and [540]n, using the formula
          given above.

          _images/rand-pascal.png

   double gsl_cdf_pascal_P(unsigned int k, double p, unsigned int n)[541]¶

   double gsl_cdf_pascal_Q(unsigned int k, double p, unsigned int n)[542]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the Pascal distribution with parameters [543]p and [544]n.

The Geometric Distribution[545]¶

   unsigned int gsl_ran_geometric(const [546]gsl_rng *r, double p)[547]¶
          This function returns a random integer from the geometric distribution,
          the number of independent trials with probability [548]p until the first
          success. The probability distribution for geometric variates is,

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

          for k \ge 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 .

   double gsl_ran_geometric_pdf(unsigned int k, double p)[549]¶
          This function computes the probability p(k) of obtaining [550]k from a
          geometric distribution with probability parameter [551]p, using the
          formula given above.

          _images/rand-geometric.png

   double gsl_cdf_geometric_P(unsigned int k, double p)[552]¶

   double gsl_cdf_geometric_Q(unsigned int k, double p)[553]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the geometric distribution with parameter [554]p.

The Hypergeometric Distribution[555]¶

   unsigned int gsl_ran_hypergeometric(const [556]gsl_rng *r, unsigned int n1,
          unsigned int n2, unsigned int t)[557]¶
          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 . The domain of k is
          \max(0, t - n_2), \ldots, \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.

   double gsl_ran_hypergeometric_pdf(unsigned int k, unsigned int n1, unsigned int
          n2, unsigned int t)[558]¶
          This function computes the probability p(k) of obtaining [559]k from a
          hypergeometric distribution with parameters [560]n1, [561]n2, [562]t,
          using the formula given above.

          _images/rand-hypergeometric.png

   double gsl_cdf_hypergeometric_P(unsigned int k, unsigned int n1, unsigned int n2,
          unsigned int t)[563]¶

   double gsl_cdf_hypergeometric_Q(unsigned int k, unsigned int n1, unsigned int n2,
          unsigned int t)[564]¶
          These functions compute the cumulative distribution functions P(k) , Q(k)
          for the hypergeometric distribution with parameters [565]n1, [566]n2 and
          [567]t.

The Logarithmic Distribution[568]¶

   unsigned int gsl_ran_logarithmic(const [569]gsl_rng *r, double p)[570]¶
          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)} {\left( p^k \over k \right)}

          for k \ge 1 .

   double gsl_ran_logarithmic_pdf(unsigned int k, double p)[571]¶
          This function computes the probability p(k) of obtaining [572]k from a
          logarithmic distribution with probability parameter [573]p, using the
          formula given above.

          _images/rand-logarithmic.png

The Wishart Distribution[574]¶

   int gsl_ran_wishart(const [575]gsl_rng *r, const double n, const [576]gsl_matrix
          *L, [577]gsl_matrix *result, [578]gsl_matrix *work)[579]¶
          This function computes a random symmetric p -by- p matrix from the Wishart
          distribution. The probability distribution for Wishart random variates is,

          p(X) = \frac{|X|^{(n-p-1)/2} e^{-\textrm{tr}\left( V^{-1}
          X\right)/2}}{2^{\frac{np}{2}} \left| V \right|^{n/2}
          \Gamma_p(\frac{n}{2})}

          Here, n > p - 1 is the number of degrees of freedom, V is a symmetric
          positive definite p -by- p scale matrix, whose Cholesky factor is
          specified by [580]L, and [581]work is p -by- p workspace. The p -by- p
          Wishart distributed matrix X is stored in [582]result on output.

   int gsl_ran_wishart_pdf(const [583]gsl_matrix *X, const [584]gsl_matrix *L_X,
          const double n, const [585]gsl_matrix *L, double *result, [586]gsl_matrix
          *work)[587]¶

   int gsl_ran_wishart_log_pdf(const [588]gsl_matrix *X, const [589]gsl_matrix *L_X,
          const double n, const [590]gsl_matrix *L, double *result, [591]gsl_matrix
          *work)[592]¶
          These functions compute p(X) or \log{p(X)} for the p -by- p matrix [593]X,
          whose Cholesky factor is specified in [594]L_X. The degrees of freedom is
          given by [595]n, the Cholesky factor of the scale matrix V is specified in
          [596]L, and [597]work is p -by- p workspace. The probably density value is
          returned in [598]result.

Shuffling and Sampling[599]¶

   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".

   void gsl_ran_shuffle(const [600]gsl_rng *r, void *base, size_t n, size_t
          size)[601]¶
          This function randomly shuffles the order of [602]n objects, each of size
          [603]size, stored in the array base[0..n-1]. The output of the random
          number generator [604]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));

   int gsl_ran_choose(const [605]gsl_rng *r, void *dest, size_t k, void *src, size_t
          n, size_t size)[606]¶
          This function fills the array dest[k] with [607]k objects taken randomly
          from the [608]n elements of the array src[0..n-1]. The objects are each of
          size [609]size. The output of the random number generator [610]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 [611]dest. It is required that [612]k be less than or equal
          to [613]n. The objects in [614]dest will be in the same relative order as
          those in [615]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));

   void gsl_ran_sample(const [616]gsl_rng *r, void *dest, size_t k, void *src,
          size_t n, size_t size)[617]¶
          This function is like [618]gsl_ran_choose() but samples [619]k items from
          the original array of [620]n items [621]src with replacement, so the same
          object can appear more than once in the output sequence [622]dest. There
          is no requirement that [623]k be less than [624]n in this case.

Examples[625]¶

   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.
#include 
#include 
#include 

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  int i, n = 10;
  double mu = 3.0;

  /* create a generator chosen by the
     environment variable GSL_RNG_TYPE */

  gsl_rng_env_setup();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  /* print n random variates chosen from
     the poisson distribution with mean
     parameter mu */

  for (i = 0; i < n; i++)
    {
      unsigned int k = gsl_ran_poisson (r, mu);
      printf (" %u", k);
    }

  printf ("\n");
  gsl_rng_free (r);
  return 0;
}

   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,
 2 5 5 2 1 0 3 4 1 1

   The variates depend on the seed used by the generator. The seed for the default
   generator type [626]gsl_rng_default can be changed with the [627]GSL_RNG_SEED
   environment variable to produce a different stream of variates:
$ GSL_RNG_SEED=123 ./a.out

   giving output
 4 5 6 3 3 1 4 2 5 5

   The following program generates a random walk in two dimensions.
#include 
#include 
#include 

int
main (void)
{
  int i;
  double x = 0, y = 0, dx, dy;

  const gsl_rng_type * T;
  gsl_rng * r;

  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  printf ("%g %g\n", x, y);

  for (i = 0; i < 10; i++)
    {
      gsl_ran_dir_2d (r, &dx, &dy);
      x += dx; y += dy;
      printf ("%g %g\n", x, y);
    }

  gsl_rng_free (r);
  return 0;
}

   [628]Fig. 5 shows the output from the program.
   [629]_images/random-walk.png

   Fig. 5 Four 10-step random walks from the origin.[630]¶

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

int
main (void)
{
  double P, Q;
  double x = 2.0;

  P = gsl_cdf_ugaussian_P (x);
  printf ("prob(x < %f) = %f\n", x, P);

  Q = gsl_cdf_ugaussian_Q (x);
  printf ("prob(x > %f) = %f\n", x, Q);

  x = gsl_cdf_ugaussian_Pinv (P);
  printf ("Pinv(%f) = %f\n", P, x);

  x = gsl_cdf_ugaussian_Qinv (Q);
  printf ("Qinv(%f) = %f\n", Q, x);

  return 0;
}

   Here is the output of the program,
prob(x < 2.000000) = 0.977250
prob(x > 2.000000) = 0.022750
Pinv(0.977250) = 2.000000
Qinv(0.022750) = 2.000000

References and Further Reading[631]¶

   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. Available online at
       [632]http://cg.scs.carleton.ca/~luc/rnbookindex.html.

   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) [633]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).

   [634]Next [635]Previous
     ____________________________________________________________________________

   © Copyright 1996-2024 The GSL Team.
   Built with [636]Sphinx using a [637]theme provided by [638]Read the Docs.

References

   Visible links:
   1. http://www.gnu.org/software/gsl/doc/html/genindex.html
   2. http://www.gnu.org/software/gsl/doc/html/search.html
   3. http://www.gnu.org/software/gsl/doc/html/statistics.html
   4. http://www.gnu.org/software/gsl/doc/html/qrng.html
   5. http://www.gnu.org/software/gsl/doc/html/index.html
   6. http://www.gnu.org/software/gsl/doc/html/intro.html
   7. http://www.gnu.org/software/gsl/doc/html/usage.html
   8. http://www.gnu.org/software/gsl/doc/html/err.html
   9. http://www.gnu.org/software/gsl/doc/html/math.html
  10. http://www.gnu.org/software/gsl/doc/html/complex.html
  11. http://www.gnu.org/software/gsl/doc/html/poly.html
  12. http://www.gnu.org/software/gsl/doc/html/specfunc.html
  13. http://www.gnu.org/software/gsl/doc/html/vectors.html
  14. http://www.gnu.org/software/gsl/doc/html/permutation.html
  15. http://www.gnu.org/software/gsl/doc/html/combination.html
  16. http://www.gnu.org/software/gsl/doc/html/multiset.html
  17. http://www.gnu.org/software/gsl/doc/html/sort.html
  18. http://www.gnu.org/software/gsl/doc/html/blas.html
  19. http://www.gnu.org/software/gsl/doc/html/linalg.html
  20. http://www.gnu.org/software/gsl/doc/html/eigen.html
  21. http://www.gnu.org/software/gsl/doc/html/fft.html
  22. http://www.gnu.org/software/gsl/doc/html/integration.html
  23. http://www.gnu.org/software/gsl/doc/html/rng.html
  24. http://www.gnu.org/software/gsl/doc/html/qrng.html
  25. http://www.gnu.org/software/gsl/doc/html/randist.html
  26. http://www.gnu.org/software/gsl/doc/html/randist.html#introduction
  27. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gaussian-distribution
  28. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gaussian-tail-distribution
  29. http://www.gnu.org/software/gsl/doc/html/randist.html#the-bivariate-gaussian-distribution
  30. http://www.gnu.org/software/gsl/doc/html/randist.html#the-multivariate-gaussian-distribution
  31. http://www.gnu.org/software/gsl/doc/html/randist.html#the-exponential-distribution
  32. http://www.gnu.org/software/gsl/doc/html/randist.html#the-laplace-distribution
  33. http://www.gnu.org/software/gsl/doc/html/randist.html#the-exponential-power-distribution
  34. http://www.gnu.org/software/gsl/doc/html/randist.html#the-cauchy-distribution
  35. http://www.gnu.org/software/gsl/doc/html/randist.html#the-rayleigh-distribution
  36. http://www.gnu.org/software/gsl/doc/html/randist.html#the-rayleigh-tail-distribution
  37. http://www.gnu.org/software/gsl/doc/html/randist.html#the-landau-distribution
  38. http://www.gnu.org/software/gsl/doc/html/randist.html#the-levy-alpha-stable-distributions
  39. http://www.gnu.org/software/gsl/doc/html/randist.html#the-levy-skew-alpha-stable-distribution
  40. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gamma-distribution
  41. http://www.gnu.org/software/gsl/doc/html/randist.html#the-flat-uniform-distribution
  42. http://www.gnu.org/software/gsl/doc/html/randist.html#the-lognormal-distribution
  43. http://www.gnu.org/software/gsl/doc/html/randist.html#the-chi-squared-distribution
  44. http://www.gnu.org/software/gsl/doc/html/randist.html#the-f-distribution
  45. http://www.gnu.org/software/gsl/doc/html/randist.html#the-t-distribution
  46. http://www.gnu.org/software/gsl/doc/html/randist.html#the-beta-distribution
  47. http://www.gnu.org/software/gsl/doc/html/randist.html#the-logistic-distribution
  48. http://www.gnu.org/software/gsl/doc/html/randist.html#the-pareto-distribution
  49. http://www.gnu.org/software/gsl/doc/html/randist.html#spherical-vector-distributions
  50. http://www.gnu.org/software/gsl/doc/html/randist.html#the-weibull-distribution
  51. http://www.gnu.org/software/gsl/doc/html/randist.html#the-type-1-gumbel-distribution
  52. http://www.gnu.org/software/gsl/doc/html/randist.html#the-type-2-gumbel-distribution
  53. http://www.gnu.org/software/gsl/doc/html/randist.html#the-dirichlet-distribution
  54. http://www.gnu.org/software/gsl/doc/html/randist.html#general-discrete-distributions
  55. http://www.gnu.org/software/gsl/doc/html/randist.html#the-poisson-distribution
  56. http://www.gnu.org/software/gsl/doc/html/randist.html#the-bernoulli-distribution
  57. http://www.gnu.org/software/gsl/doc/html/randist.html#the-binomial-distribution
  58. http://www.gnu.org/software/gsl/doc/html/randist.html#the-multinomial-distribution
  59. http://www.gnu.org/software/gsl/doc/html/randist.html#the-negative-binomial-distribution
  60. http://www.gnu.org/software/gsl/doc/html/randist.html#the-pascal-distribution
  61. http://www.gnu.org/software/gsl/doc/html/randist.html#the-geometric-distribution
  62. http://www.gnu.org/software/gsl/doc/html/randist.html#the-hypergeometric-distribution
  63. http://www.gnu.org/software/gsl/doc/html/randist.html#the-logarithmic-distribution
  64. http://www.gnu.org/software/gsl/doc/html/randist.html#the-wishart-distribution
  65. http://www.gnu.org/software/gsl/doc/html/randist.html#shuffling-and-sampling
  66. http://www.gnu.org/software/gsl/doc/html/randist.html#examples
  67. http://www.gnu.org/software/gsl/doc/html/randist.html#references-and-further-reading
  68. http://www.gnu.org/software/gsl/doc/html/statistics.html
  69. http://www.gnu.org/software/gsl/doc/html/rstat.html
  70. http://www.gnu.org/software/gsl/doc/html/movstat.html
  71. http://www.gnu.org/software/gsl/doc/html/filter.html
  72. http://www.gnu.org/software/gsl/doc/html/histogram.html
  73. http://www.gnu.org/software/gsl/doc/html/ntuple.html
  74. http://www.gnu.org/software/gsl/doc/html/montecarlo.html
  75. http://www.gnu.org/software/gsl/doc/html/siman.html
  76. http://www.gnu.org/software/gsl/doc/html/ode-initval.html
  77. http://www.gnu.org/software/gsl/doc/html/interp.html
  78. http://www.gnu.org/software/gsl/doc/html/diff.html
  79. http://www.gnu.org/software/gsl/doc/html/cheb.html
  80. http://www.gnu.org/software/gsl/doc/html/sum.html
  81. http://www.gnu.org/software/gsl/doc/html/dwt.html
  82. http://www.gnu.org/software/gsl/doc/html/dht.html
  83. http://www.gnu.org/software/gsl/doc/html/roots.html
  84. http://www.gnu.org/software/gsl/doc/html/min.html
  85. http://www.gnu.org/software/gsl/doc/html/multiroots.html
  86. http://www.gnu.org/software/gsl/doc/html/multimin.html
  87. http://www.gnu.org/software/gsl/doc/html/lls.html
  88. http://www.gnu.org/software/gsl/doc/html/nls.html
  89. http://www.gnu.org/software/gsl/doc/html/bspline.html
  90. http://www.gnu.org/software/gsl/doc/html/spmatrix.html
  91. http://www.gnu.org/software/gsl/doc/html/spblas.html
  92. http://www.gnu.org/software/gsl/doc/html/splinalg.html
  93. http://www.gnu.org/software/gsl/doc/html/const.html
  94. http://www.gnu.org/software/gsl/doc/html/ieee754.html
  95. http://www.gnu.org/software/gsl/doc/html/debug.html
  96. http://www.gnu.org/software/gsl/doc/html/contrib.html
  97. http://www.gnu.org/software/gsl/doc/html/autoconf.html
  98. http://www.gnu.org/software/gsl/doc/html/cblas.html
  99. http://www.gnu.org/software/gsl/doc/html/gpl.html
 100. http://www.gnu.org/software/gsl/doc/html/fdl.html
 101. http://www.gnu.org/software/gsl/doc/html/index.html
 102. http://www.gnu.org/software/gsl/doc/html/_sources/randist.rst.txt
 103. http://www.gnu.org/software/gsl/doc/html/statistics.html
 104. http://www.gnu.org/software/gsl/doc/html/qrng.html
 105. http://www.gnu.org/software/gsl/doc/html/randist.html#random-number-distributions
 106. http://www.gnu.org/software/gsl/doc/html/randist.html#introduction
 107. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gaussian-distribution
 108. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 109. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian
 110. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian
 111. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian
 112. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian
 113. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_pdf
 114. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_pdf
 115. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_pdf
 116. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 117. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_ziggurat
 118. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 119. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_ratio_method
 120. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 121. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_ugaussian
 122. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_ugaussian_pdf
 123. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 124. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_ugaussian_ratio_method
 125. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gaussian_P
 126. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gaussian_Q
 127. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gaussian_Pinv
 128. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gaussian_Qinv
 129. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gaussian_Qinv
 130. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_ugaussian_P
 131. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_ugaussian_Q
 132. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_ugaussian_Pinv
 133. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_ugaussian_Qinv
 134. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gaussian-tail-distribution
 135. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 136. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail
 137. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail
 138. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail
 139. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail_pdf
 140. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail_pdf
 141. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail_pdf
 142. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gaussian_tail_pdf
 143. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 144. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_ugaussian_tail
 145. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_ugaussian_tail_pdf
 146. http://www.gnu.org/software/gsl/doc/html/randist.html#the-bivariate-gaussian-distribution
 147. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 148. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian
 149. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian
 150. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian
 151. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian
 152. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian
 153. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 154. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 155. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 156. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 157. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 158. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bivariate_gaussian_pdf
 159. http://www.gnu.org/software/gsl/doc/html/randist.html#the-multivariate-gaussian-distribution
 160. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 161. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 162. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 163. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 164. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian
 165. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian
 166. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian
 167. http://www.gnu.org/software/gsl/doc/html/linalg.html#c.gsl_linalg_cholesky_decomp
 168. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian
 169. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 170. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 171. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 172. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 173. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_pdf
 174. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 175. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 176. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 177. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 178. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_log_pdf
 179. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_log_pdf
 180. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_log_pdf
 181. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_log_pdf
 182. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_log_pdf
 183. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 184. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
 185. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_mean
 186. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_mean
 187. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_mean
 188. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 189. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 190. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_vcov
 191. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_vcov
 192. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multivariate_gaussian_vcov
 193. http://www.gnu.org/software/gsl/doc/html/randist.html#the-exponential-distribution
 194. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 195. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exponential
 196. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exponential
 197. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exponential_pdf
 198. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exponential_pdf
 199. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exponential_pdf
 200. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exponential_P
 201. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exponential_Q
 202. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exponential_Pinv
 203. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exponential_Qinv
 204. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exponential_Qinv
 205. http://www.gnu.org/software/gsl/doc/html/randist.html#the-laplace-distribution
 206. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 207. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_laplace
 208. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_laplace
 209. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_laplace_pdf
 210. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_laplace_pdf
 211. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_laplace_pdf
 212. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_laplace_P
 213. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_laplace_Q
 214. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_laplace_Pinv
 215. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_laplace_Qinv
 216. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_laplace_Qinv
 217. http://www.gnu.org/software/gsl/doc/html/randist.html#the-exponential-power-distribution
 218. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 219. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow
 220. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow
 221. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow
 222. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow_pdf
 223. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow_pdf
 224. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow_pdf
 225. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_exppow_pdf
 226. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exppow_P
 227. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exppow_Q
 228. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exppow_Q
 229. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_exppow_Q
 230. http://www.gnu.org/software/gsl/doc/html/randist.html#the-cauchy-distribution
 231. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 232. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_cauchy
 233. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_cauchy
 234. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_cauchy_pdf
 235. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_cauchy_pdf
 236. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_cauchy_pdf
 237. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_cauchy_P
 238. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_cauchy_Q
 239. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_cauchy_Pinv
 240. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_cauchy_Qinv
 241. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_cauchy_Qinv
 242. http://www.gnu.org/software/gsl/doc/html/randist.html#the-rayleigh-distribution
 243. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 244. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh
 245. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh
 246. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_pdf
 247. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_pdf
 248. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_pdf
 249. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_rayleigh_P
 250. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_rayleigh_Q
 251. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_rayleigh_Pinv
 252. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_rayleigh_Qinv
 253. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_rayleigh_Qinv
 254. http://www.gnu.org/software/gsl/doc/html/randist.html#the-rayleigh-tail-distribution
 255. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 256. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail
 257. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail
 258. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail
 259. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail_pdf
 260. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail_pdf
 261. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail_pdf
 262. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_rayleigh_tail_pdf
 263. http://www.gnu.org/software/gsl/doc/html/randist.html#the-landau-distribution
 264. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 265. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_landau
 266. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_landau_pdf
 267. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_landau_pdf
 268. http://www.gnu.org/software/gsl/doc/html/randist.html#the-levy-alpha-stable-distributions
 269. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 270. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy
 271. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy
 272. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy
 273. http://www.gnu.org/software/gsl/doc/html/randist.html#the-levy-skew-alpha-stable-distribution
 274. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 275. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy_skew
 276. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy_skew
 277. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy_skew
 278. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_levy_skew
 279. http://www.gnu.org/software/gsl/doc/html/randist.html#the-gamma-distribution
 280. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 281. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma
 282. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma
 283. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 284. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma_knuth
 285. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma_pdf
 286. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma_pdf
 287. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma_pdf
 288. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gamma_pdf
 289. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_P
 290. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_Q
 291. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_Pinv
 292. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_Qinv
 293. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_Qinv
 294. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gamma_Qinv
 295. http://www.gnu.org/software/gsl/doc/html/randist.html#the-flat-uniform-distribution
 296. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 297. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat
 298. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat
 299. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat
 300. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat_pdf
 301. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat_pdf
 302. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat_pdf
 303. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_flat_pdf
 304. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_P
 305. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_Q
 306. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_Pinv
 307. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_Qinv
 308. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_Qinv
 309. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_flat_Qinv
 310. http://www.gnu.org/software/gsl/doc/html/randist.html#the-lognormal-distribution
 311. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 312. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_lognormal
 313. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_lognormal_pdf
 314. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_lognormal_pdf
 315. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_lognormal_pdf
 316. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_lognormal_pdf
 317. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_P
 318. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_Q
 319. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_Pinv
 320. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_Qinv
 321. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_Qinv
 322. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_lognormal_Qinv
 323. http://www.gnu.org/software/gsl/doc/html/randist.html#the-chi-squared-distribution
 324. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 325. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_chisq
 326. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_chisq
 327. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_chisq_pdf
 328. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_chisq_pdf
 329. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_chisq_pdf
 330. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_chisq_P
 331. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_chisq_Q
 332. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_chisq_Pinv
 333. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_chisq_Qinv
 334. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_chisq_Qinv
 335. http://www.gnu.org/software/gsl/doc/html/randist.html#the-f-distribution
 336. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 337. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist
 338. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist
 339. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist
 340. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist_pdf
 341. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist_pdf
 342. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist_pdf
 343. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_fdist_pdf
 344. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_P
 345. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_Q
 346. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_Pinv
 347. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_Qinv
 348. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_Qinv
 349. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_fdist_Qinv
 350. http://www.gnu.org/software/gsl/doc/html/randist.html#the-t-distribution
 351. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 352. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_tdist
 353. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_tdist_pdf
 354. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_tdist_pdf
 355. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_tdist_pdf
 356. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_tdist_P
 357. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_tdist_Q
 358. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_tdist_Pinv
 359. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_tdist_Qinv
 360. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_tdist_Qinv
 361. http://www.gnu.org/software/gsl/doc/html/randist.html#the-beta-distribution
 362. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 363. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_beta
 364. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_beta_pdf
 365. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_beta_pdf
 366. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_beta_pdf
 367. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_beta_pdf
 368. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_P
 369. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_Q
 370. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_Pinv
 371. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_Qinv
 372. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_Qinv
 373. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_beta_Qinv
 374. http://www.gnu.org/software/gsl/doc/html/randist.html#the-logistic-distribution
 375. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 376. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logistic
 377. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logistic_pdf
 378. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logistic_pdf
 379. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logistic_pdf
 380. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_logistic_P
 381. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_logistic_Q
 382. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_logistic_Pinv
 383. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_logistic_Qinv
 384. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_logistic_Qinv
 385. http://www.gnu.org/software/gsl/doc/html/randist.html#the-pareto-distribution
 386. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 387. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto
 388. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto
 389. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto_pdf
 390. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto_pdf
 391. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto_pdf
 392. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pareto_pdf
 393. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_P
 394. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_Q
 395. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_Pinv
 396. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_Qinv
 397. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_Qinv
 398. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pareto_Qinv
 399. http://www.gnu.org/software/gsl/doc/html/randist.html#spherical-vector-distributions
 400. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 401. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d
 402. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 403. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 404. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 405. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 406. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 407. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 408. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 409. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_2d_trig_method
 410. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 411. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_3d
 412. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_3d
 413. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_3d
 414. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_3d
 415. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 416. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_nd
 417. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dir_nd
 418. http://www.gnu.org/software/gsl/doc/html/randist.html#the-weibull-distribution
 419. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 420. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_weibull
 421. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_weibull_pdf
 422. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_weibull_pdf
 423. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_weibull_pdf
 424. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_weibull_pdf
 425. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_P
 426. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_Q
 427. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_Pinv
 428. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_Qinv
 429. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_Qinv
 430. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_weibull_Qinv
 431. http://www.gnu.org/software/gsl/doc/html/randist.html#the-type-1-gumbel-distribution
 432. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 433. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel1
 434. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel1_pdf
 435. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel1_pdf
 436. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel1_pdf
 437. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel1_pdf
 438. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_P
 439. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_Q
 440. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_Pinv
 441. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_Qinv
 442. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_Qinv
 443. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel1_Qinv
 444. http://www.gnu.org/software/gsl/doc/html/randist.html#the-type-2-gumbel-distribution
 445. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 446. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel2
 447. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel2_pdf
 448. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel2_pdf
 449. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel2_pdf
 450. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_gumbel2_pdf
 451. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_P
 452. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_Q
 453. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_Pinv
 454. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_Qinv
 455. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_Qinv
 456. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_gumbel2_Qinv
 457. http://www.gnu.org/software/gsl/doc/html/randist.html#the-dirichlet-distribution
 458. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 459. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet
 460. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet
 461. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet
 462. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet
 463. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet_pdf
 464. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_dirichlet_lnpdf
 465. http://www.gnu.org/software/gsl/doc/html/randist.html#general-discrete-distributions
 466. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_t
 467. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_t
 468. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_preproc
 469. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_preproc
 470. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete
 471. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 472. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_t
 473. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete
 474. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_t
 475. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_pdf
 476. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_pdf
 477. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_t
 478. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_free
 479. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_discrete_free
 480. http://www.gnu.org/software/gsl/doc/html/randist.html#the-poisson-distribution
 481. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 482. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_poisson
 483. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_poisson
 484. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_poisson_pdf
 485. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_poisson_pdf
 486. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_poisson_pdf
 487. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_poisson_P
 488. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_poisson_Q
 489. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_poisson_Q
 490. http://www.gnu.org/software/gsl/doc/html/randist.html#the-bernoulli-distribution
 491. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 492. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bernoulli
 493. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bernoulli
 494. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bernoulli_pdf
 495. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bernoulli_pdf
 496. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_bernoulli_pdf
 497. http://www.gnu.org/software/gsl/doc/html/randist.html#the-binomial-distribution
 498. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 499. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial
 500. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial
 501. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial
 502. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial_pdf
 503. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial_pdf
 504. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial_pdf
 505. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_binomial_pdf
 506. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_binomial_P
 507. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_binomial_Q
 508. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_binomial_Q
 509. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_binomial_Q
 510. http://www.gnu.org/software/gsl/doc/html/randist.html#the-multinomial-distribution
 511. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 512. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 513. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 514. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 515. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 516. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 517. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 518. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial
 519. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial_pdf
 520. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_multinomial_lnpdf
 521. http://www.gnu.org/software/gsl/doc/html/randist.html#the-negative-binomial-distribution
 522. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 523. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial
 524. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial
 525. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial
 526. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial_pdf
 527. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial_pdf
 528. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial_pdf
 529. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_negative_binomial_pdf
 530. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_negative_binomial_P
 531. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_negative_binomial_Q
 532. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_negative_binomial_Q
 533. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_negative_binomial_Q
 534. http://www.gnu.org/software/gsl/doc/html/randist.html#the-pascal-distribution
 535. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 536. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pascal
 537. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pascal_pdf
 538. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pascal_pdf
 539. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pascal_pdf
 540. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_pascal_pdf
 541. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pascal_P
 542. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pascal_Q
 543. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pascal_Q
 544. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_pascal_Q
 545. http://www.gnu.org/software/gsl/doc/html/randist.html#the-geometric-distribution
 546. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 547. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_geometric
 548. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_geometric
 549. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_geometric_pdf
 550. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_geometric_pdf
 551. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_geometric_pdf
 552. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_geometric_P
 553. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_geometric_Q
 554. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_geometric_Q
 555. http://www.gnu.org/software/gsl/doc/html/randist.html#the-hypergeometric-distribution
 556. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 557. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric
 558. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric_pdf
 559. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric_pdf
 560. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric_pdf
 561. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric_pdf
 562. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_hypergeometric_pdf
 563. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_hypergeometric_P
 564. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_hypergeometric_Q
 565. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_hypergeometric_Q
 566. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_hypergeometric_Q
 567. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_cdf_hypergeometric_Q
 568. http://www.gnu.org/software/gsl/doc/html/randist.html#the-logarithmic-distribution
 569. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 570. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logarithmic
 571. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logarithmic_pdf
 572. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logarithmic_pdf
 573. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_logarithmic_pdf
 574. http://www.gnu.org/software/gsl/doc/html/randist.html#the-wishart-distribution
 575. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 576. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 577. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 578. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 579. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart
 580. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart
 581. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart
 582. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart
 583. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 584. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 585. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 586. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 587. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_pdf
 588. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 589. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 590. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 591. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
 592. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 593. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 594. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 595. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 596. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 597. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 598. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_wishart_log_pdf
 599. http://www.gnu.org/software/gsl/doc/html/randist.html#shuffling-and-sampling
 600. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 601. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_shuffle
 602. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_shuffle
 603. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_shuffle
 604. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_shuffle
 605. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 606. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 607. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 608. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 609. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 610. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 611. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 612. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 613. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 614. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 615. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 616. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng
 617. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 618. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_choose
 619. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 620. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 621. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 622. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 623. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 624. http://www.gnu.org/software/gsl/doc/html/randist.html#c.gsl_ran_sample
 625. http://www.gnu.org/software/gsl/doc/html/randist.html#examples
 626. http://www.gnu.org/software/gsl/doc/html/rng.html#c.gsl_rng_default
 627. http://www.gnu.org/software/gsl/doc/html/rng.html#c.GSL_RNG_SEED
 628. http://www.gnu.org/software/gsl/doc/html/randist.html#fig-rand-walk
 629. http://www.gnu.org/software/gsl/doc/html/_images/random-walk.png
 630. http://www.gnu.org/software/gsl/doc/html/randist.html#id1
 631. http://www.gnu.org/software/gsl/doc/html/randist.html#references-and-further-reading
 632. http://cg.scs.carleton.ca/~luc/rnbookindex.html
 633. http://pdg.lbl.gov/
 634. http://www.gnu.org/software/gsl/doc/html/statistics.html
 635. http://www.gnu.org/software/gsl/doc/html/qrng.html
 636. https://www.sphinx-doc.org/
 637. https://github.com/readthedocs/sphinx_rtd_theme
 638. https://readthedocs.org/

   Hidden links:
 640. http://www.gnu.org/software/gsl/doc/html/index.html