randist.texi

开放gsl矩阵运算

@cindex random number distributions

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
@dfn{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 functions described in this section are declared in
@file{gsl_randist.h}.

@menu
* The Gaussian Distribution::   
* The Gaussian Tail Distribution::  
* The Bivariate Gaussian Distribution::  
* The Exponential Distribution::  
* The Laplace Distribution::    
* The Exponential Power Distribution::  
* The Cauchy Distribution::     
* The Rayleigh Distribution::   
* The Rayleigh Tail Distribution::  
* The Landau Distribution::     
* The Levy alpha-Stable Distributions::  
* The Levy skew alpha-Stable Distribution::  
* The Gamma Distribution::      
* The Flat (Uniform) Distribution::  
* The Lognormal Distribution::  
* The Chi-squared Distribution::  
* The F-distribution::          
* The t-distribution::          
* The Beta Distribution::       
* The Logistic Distribution::   
* The Pareto Distribution::     
* The Spherical Distribution (2D & 3D)::  
* The Weibull Distribution::    
* The Type-1 Gumbel Distribution::  
* The Type-2 Gumbel Distribution::  
* General Discrete Distributions::  
* The Poisson Distribution::    
* The Bernoulli Distribution::  
* The Binomial Distribution::   
* The Negative Binomial Distribution::  
* The Pascal Distribution::     
* The Geometric Distribution::  
* The Hypergeometric Distribution::  
* The Logarithmic Distribution::  
* Shuffling and Sampling::      
* Random Number Distribution Examples::  
* Random Number Distribution References and Further Reading::  
@end menu

@page
@node The Gaussian Distribution
@section The Gaussian Distribution
@deftypefn Random double gsl_ran_gaussian (const gsl_rng * @var{r}, double @var{sigma})
@cindex Gaussian random variates
This function returns a Gaussian random variate, with mean zero and
standard deviation @var{sigma}.  The probability distribution for
Gaussian random variates is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over \sqrt@{2 \pi \sigma^2@}@} \exp (-x^2 / 2\sigma^2) dx
@end example
@end ifinfo
@noindent
for @math{x} in the range @math{-\infty} to @math{+\infty}.  Use the
transformation @math{z = \mu + x} on the numbers returned by
@code{gsl_ran_gaussian} to obtain a Gaussian distribution with mean
@math{\mu}.  This function uses the Box-Mueller algorithm which requires two
calls the random number generator @var{r}.
@end deftypefn

@deftypefun double gsl_ran_gaussian_pdf (double @var{x}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a Gaussian distribution with standard deviation @var{sigma}, using
the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-gaussian.tex}
@end tex

@deftypefun double gsl_ran_gaussian_ratio_method (const gsl_rng * @var{r}, const double @var{sigma})
This function computes a gaussian random variate using the
Kinderman-Monahan ratio method.
@end deftypefun

@deftypefn Random double gsl_ran_ugaussian (const gsl_rng * @var{r})
@deftypefnx Random double gsl_ran_ugaussian_pdf (double @var{x})
@deftypefnx Random double gsl_ran_ugaussian_ratio_method (const gsl_rng * @var{r}, const double @var{sigma})
These functions compute results for the unit Gaussian distribution.  They
are equivalent to the functions above with a standard deviation of one,
@var{sigma} = 1.
@end deftypefn

@page
@node The Gaussian Tail Distribution
@section The Gaussian Tail Distribution
@deftypefn Random double gsl_ran_gaussian_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma})
@cindex Gaussian Tail random variates
This function provides random variates from the upper tail of a Gaussian
distribution with standard deviation @var{sigma}.  The values returned
are larger than the lower limit @var{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,

@tex
\beforedisplay
$$
p(x) dx = {1 \over N(a;\sigma)} \exp (- x^2 / 2\sigma^2) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over N(a;\sigma)@} \exp (- x^2/(2 \sigma^2)) dx
@end example
@end ifinfo

@noindent
for @math{x > a} where @math{N(a;\sigma)} is the normalization constant,

@tex
\beforedisplay
$$
N(a;\sigma) = {1 \over 2} \hbox{erfc}\left({a \over \sqrt{2 \sigma^2}}\right).
$$
\afterdisplay
@end tex
@ifinfo
@example
N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
@end example
@end ifinfo

@end deftypefn

@deftypefun double gsl_ran_gaussian_tail_pdf (double @var{x}, double @var{a}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a Gaussian tail distribution with standard deviation @var{sigma} and
lower limit @var{a}, using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-gaussian-tail.tex}
@end tex

@deftypefn Random double gsl_ran_ugaussian_tail (const gsl_rng * @var{r}, double @var{a})
@deftypefnx Random double gsl_ran_ugaussian_tail_pdf (double @var{x}, double @var{a})
These functions compute results for the tail of a unit Gaussian
distribution.  They are equivalent to the functions above with a standard
deviation of one, @var{sigma} = 1.
@end deftypefn


@page
@node The Bivariate Gaussian Distribution
@section The Bivariate Gaussian Distribution

@deftypefn Random void gsl_ran_bivariate_gaussian (const gsl_rng * @var{r}, double @var{sigma_x}, double @var{sigma_y}, double @var{rho}, double * @var{x}, double * @var{y})
@cindex Gaussian random variates
This function generates a pair of correlated gaussian variates, with
mean zero, correlation coefficient @var{rho} and standard deviations
@var{sigma_x} and @var{sigma_y} in the @math{x} and @math{y} directions.
The probability distribution for bivariate gaussian random variates is,

@tex
\beforedisplay
$$
p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left(-(x^2 + y^2 - 2 \rho x y) \over 2\sigma_x^2\sigma_y^2 (1-\rho^2)\right) dx dy
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x,y) dx dy = @{1 \over 2 \pi \sigma_x \sigma_y \sqrt@{1-\rho^2@}@} \exp (-(x^2 + y^2 - 2 \rho x y)/2\sigma_x^2\sigma_y^2 (1-\rho^2)) dx dy
@end example
@end ifinfo

@noindent
for @math{x,y} in the range @math{-\infty} to @math{+\infty}.  The
correlation coefficient @var{rho} should lie between @math{1} and
@math{-1}.
@end deftypefn

@deftypefun double gsl_ran_bivariate_gaussian_pdf (double @var{x}, double @var{y}, double @var{sigma_x}, double @var{sigma_y}, double @var{rho})
This function computes the probability density @math{p(x,y)} at
(@var{x},@var{y}) for a bivariate gaussian distribution with standard
deviations @var{sigma_x}, @var{sigma_y} and correlation coefficient
@var{rho}, using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-bivariate-gaussian.tex}
@end tex

@page
@node The Exponential Distribution
@section The Exponential Distribution
@deftypefn Random double gsl_ran_exponential (const gsl_rng * @var{r}, double @var{mu})
@cindex Exponential random variates
This function returns a random variate from the exponential distribution
with mean @var{mu}. The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over \mu@} \exp(-x/\mu) dx
@end example
@end ifinfo

@noindent
for @c{$x \ge 0$}
@math{x >= 0}. 
@end deftypefn

@deftypefun double gsl_ran_exponential_pdf (double @var{x}, double @var{mu})
This function computes the probability density @math{p(x)} at @var{x}
for an exponential distribution with mean @var{mu}, using the formula
given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-exponential.tex}
@end tex

@page
@node The Laplace Distribution
@section The Laplace Distribution
@deftypefn Random double gsl_ran_laplace (const gsl_rng * @var{r}, double @var{a})
@cindex two-sided exponential random variates
@cindex Laplace distribution random variates
This function returns a random variate from the the Laplace distribution
with width @var{a}.  The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over 2 a@}  \exp(-|x/a|) dx
@end example
@end ifinfo

@noindent
for @math{-\infty < x < \infty}.
@end deftypefn

@deftypefun double gsl_ran_laplace_pdf (double @var{x}, double @var{a})
This function computes the probability density @math{p(x)} at @var{x}
for a Laplace distribution with mean @var{a}, using the formula
given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-laplace.tex}
@end tex

@page
@node The Exponential Power Distribution
@section The Exponential Power Distribution
@deftypefn Random double gsl_ran_exppow (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex Exponential power distribution, random variates
This function returns a random variate from the exponential power distribution
with scale parameter @var{a} and exponent @var{b}.  The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over 2 a \Gamma(1+1/b)@} \exp(-|x/a|^b) dx
@end example
@end ifinfo
@noindent
for @c{$x \ge 0$}
@math{x >= 0}.  For @math{b = 1} this reduces to the Laplace
distribution.  For @math{b = 2} it has the same form as a gaussian
distribution, but with @c{$a = \sqrt{2} \sigma$}
@math{a = \sqrt@{2@} \sigma}.
@end deftypefn

@deftypefun double gsl_ran_exppow_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for an exponential power distribution with scale parameter @var{a}
and exponent @var{b}, using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-exppow.tex}
@end tex

@page
@node The Cauchy Distribution
@section The Cauchy Distribution
@deftypefn Random double gsl_ran_cauchy (const gsl_rng * @var{r}, double @var{a})
@cindex Cauchy random variates
This function returns a random variate from the Cauchy distribution with
scale parameter @var{a}.  The probability distribution for Cauchy
random variates is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over a\pi (1 + (x/a)^2) @} dx
@end example
@end ifinfo

@noindent
for @math{x} in the range @math{-\infty} to @math{+\infty}.  The Cauchy
distribution is also known as the Lorentz distribution.
@end deftypefn

@deftypefun double gsl_ran_cauchy_pdf (double @var{x}, double @var{a})
This function computes the probability density @math{p(x)} at @var{x}
for a Cauchy distribution with scale parameter @var{a}, using the formula
given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-cauchy.tex}
@end tex

@page
@node The Rayleigh Distribution
@section The Rayleigh Distribution
@deftypefn Random double gsl_ran_rayleigh (const gsl_rng * @var{r}, double @var{sigma})
@cindex Rayleigh random variates
This function returns a random variate from the Rayleigh distribution with
scale parameter @var{sigma}.  The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{x \over \sigma^2@} \exp(- x^2/(2 \sigma^2)) dx
@end example
@end ifinfo
@noindent
for @math{x > 0}.
@end deftypefn

@deftypefun double gsl_ran_rayleigh_pdf (double @var{x}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a Rayleigh distribution with scale parameter @var{sigma}, using the
formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-rayleigh.tex}
@end tex

@page
@node The Rayleigh Tail Distribution
@section The Rayleigh Tail Distribution
@deftypefn Random double gsl_ran_rayleigh_tail (const gsl_rng * @var{r}, double @var{a} double @var{sigma})
@cindex Rayleigh Tail random variates
This function returns a random variate from the tail of the Rayleigh
distribution with scale parameter @var{sigma} and a lower limit of
@var{a}.  The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{x \over \sigma^2@} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
@end example
@end ifinfo

@noindent
for @math{x > a}.
@end deftypefn

@deftypefun double gsl_ran_rayleigh_tail_pdf (double @var{x}, double @var{a}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a Rayleigh tail distribution with scale parameter @var{sigma} and
lower limit @var{a}, using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-rayleigh-tail.tex}
@end tex

@page
@node The Landau Distribution
@section The Landau Distribution
@deftypefn Random double gsl_ran_landau (const gsl_rng * @var{r})
@cindex Landau random variates