randist.texi

该文件为c++的数学函数库!是一个非常有用的编程工具.它含有各种数学函数,为科学计算、工程应用等程序编写提供方便！

@cindex random number distributions
@cindex cumulative distribution functions (CDFs)
@cindex CDFs, cumulative distribution functions
@cindex inverse cumulative distribution functions
@cindex quantile functions
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 library also provides cumulative distribution functions and inverse
cumulative distribution functions, sometimes referred to as quantile
functions.  The cumulative distribution functions and their inverses are
computed separately for the upper and lower tails of the distribution,
allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions
described in this section are declared in @file{gsl_randist.h}.  The
corresponding cumulative distribution functions are declared in
@file{gsl_cdf.h}.

@menu
* Random Number Distribution Introduction::  
* 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::  
* The Dirichlet Distribution::  
* General Discrete Distributions::  
* The Poisson Distribution::    
* The Bernoulli Distribution::  
* The Binomial Distribution::
* The Multinomial 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

@node Random Number Distribution Introduction
@section Introduction

Continuous random number distributions are defined by a probability
density function, @math{p(x)}, such that the probability of @math{x}
occurring in the infinitesimal range @math{x} to @math{x+dx} is @c{$p\,dx$}
@math{p dx}.

The cumulative distribution function for the lower tail is defined by,
@tex
\beforedisplay
$$
P(x) = \int_{-\infty}^{x} dx' p(x')
$$
\afterdisplay
@end tex
@ifinfo
@example
P(x) = \int_@{-\infty@}^@{x@} dx' p(x')
@end example
@end ifinfo
@noindent
and gives the probability of a variate taking a value less than @math{x}.

The cumulative distribution function for the upper tail is defined by,
@tex
\beforedisplay
$$
Q(x) = \int_{x}^{-\infty} dx' p(x')
$$
\afterdisplay
@end tex
@ifinfo
@example
P(x) = \int_@{x@}^@{-\infty@} dx' p(x')
@end example
@end ifinfo
@noindent
and gives the probability of a variate taking a greater than @math{x}.
The upper and lower cumulative distribution functions are related by
@math{P(x) + Q(x) = 1} and satisfy @c{$0 \le P(x) \le 1$}
@math{0 <= P(x) <= 1}, @c{$0 \le Q(x) \le 1$}
@math{0 <= Q(x) <= 1}.

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

@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 distribution
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 to 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}, 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 Function double gsl_ran_ugaussian_pdf (double @var{x})
@deftypefnx Random double gsl_ran_ugaussian_ratio_method (const gsl_rng * @var{r})
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

@deftypefun double gsl_cdf_gaussian_P (double @var{x}, double @var{sigma})
@deftypefunx double gsl_cdf_gaussian_Q (double @var{x}, double @var{sigma})
@deftypefunx double gsl_cdf_gaussian_Pinv (double @var{P}, double @var{sigma})
@deftypefunx double gsl_cdf_gaussian_Qinv (double @var{Q}, double @var{sigma})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the Gaussian
distribution with standard deviation @var{sigma}.
@end deftypefun

@deftypefun double gsl_cdf_ugaussian_P (double @var{x})
@deftypefunx double gsl_cdf_ugaussian_Q (double @var{x})
@deftypefunx double gsl_cdf_ugaussian_Pinv (double @var{P})
@deftypefunx double gsl_cdf_ugaussian_Qinv (double @var{Q})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the unit Gaussian
distribution.
@end deftypefun

@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 distribution
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 Function 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 Bivariate Gaussian distribution
@cindex Two-dimensional Gaussian distribution
@cindex Gaussian distribution, bivariate
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
$$
p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left(-{(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y)) \over 2(1-\rho^2)}\right) dx dy
$$
\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/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\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 distribution
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

@deftypefun double gsl_cdf_exponential_P (double @var{x}, double @var{mu})
@deftypefunx double gsl_cdf_exponential_Q (double @var{x}, double @var{mu})
@deftypefunx double gsl_cdf_exponential_Pinv (double @var{P}, double @var{mu})
@deftypefunx double gsl_cdf_exponential_Qinv (double @var{Q}, double @var{mu})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the exponential
distribution with mean @var{mu}.
@end deftypefun

@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 distribution
@cindex Laplace distribution
This function returns a random variate from 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 width @var{a}, using the formula
given above.
@end deftypefun

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

@deftypefun double gsl_cdf_laplace_P (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_laplace_Q (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_laplace_Pinv (double @var{P}, double @var{a})
@deftypefunx double gsl_cdf_laplace_Qinv (double @var{Q}, double @var{a})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the Laplace
distribution with width @var{a}.
@end deftypefun


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