randist.texi

This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY without ev

@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
@deftypefun 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 deftypefun

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

@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

@deftypefun double gsl_cdf_exppow_P (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_exppow_Q (double @var{x}, double @var{a}, double @var{b})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} for the exponential power distribution with
parameters @var{a} and @var{b}.
@end deftypefun


@page
@node The Cauchy Distribution
@section The Cauchy Distribution
@deftypefun double gsl_ran_cauchy (const gsl_rng * @var{r}, double @var{a})
@cindex Cauchy distribution
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 deftypefun

@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

@deftypefun double gsl_cdf_cauchy_P (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_cauchy_Q (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_cauchy_Pinv (double @var{P}, double @var{a})
@deftypefunx double gsl_cdf_cauchy_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 Cauchy
distribution with scale parameter @var{a}.
@end deftypefun


@page
@node The Rayleigh Distribution
@section The Rayleigh Distribution
@deftypefun double gsl_ran_rayleigh (const gsl_rng * @var{r}, double @var{sigma})
@cindex Rayleigh distribution
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 deftypefun

@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

@deftypefun double gsl_cdf_rayleigh_P (double @var{x}, double @var{sigma})
@deftypefunx double gsl_cdf_rayleigh_Q (double @var{x}, double @var{sigma})
@deftypefunx double gsl_cdf_rayleigh_Pinv (double @var{P}, double @var{sigma})
@deftypefunx double gsl_cdf_rayleigh_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 Rayleigh
distribution with scale parameter @var{sigma}.
@end deftypefun


@page
@node The Rayleigh Tail Distribution
@section The Rayleigh Tail Distribution
@deftypefun double gsl_ran_rayleigh_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma})
@cindex Rayleigh Tail distribution
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 deftypefun

@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
@deftypefun double gsl_ran_landau (const gsl_rng * @var{r})
@cindex Landau distribution
This function returns a random variate from the Landau distribution.  The
probability distribution for Landau random variates is defined
analytically by the complex integral,
@tex
\beforedisplay
$$
p(x) = 
{1 \over {2 \pi i}} \int_{c-i\infty}^{c+i\infty} ds\, \exp(s \log(s) + x s) 
$$
\afterdisplay
@end tex
@ifinfo

@example
p(x) = (1/(2 \pi i)) \int_@{c-i\infty@}^@{c+i\infty@} ds exp(s log(s) + x s) 
@end example
@end ifinfo
For numerical purposes it is more convenient to use the following
equivalent form of the integral,
@tex
\beforedisplay
$$
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
$$
\afterdisplay
@end tex
@ifinfo

@example
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
@end example
@end ifinfo
@end deftypefun

@deftypefun double gsl_ran_landau_pdf (double @var{x})
This function computes the probability density @math{p(x)} at @var{x}
for the Landau distribution using an approximation to the formula given
above.
@end deftypefun

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

@page
@node The Levy alpha-Stable Distributions
@section The Levy alpha-Stable Distributions
@deftypefun double gsl_ran_levy (const gsl_rng * @var{r}, double @var{c}, double @var{alpha})
@cindex Levy distribution
This function returns a random variate from the Levy symmetric stable
distribution with scale @var{c} and exponent @var{alpha}.  The symmetric
stable probability distribution is defined by a fourier transform,
@tex
\beforedisplay
$$
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha)
$$
\afterdisplay
@end tex
@ifinfo

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

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

The algorithm only works for @c{$0 < \alpha \le 2$}
@math{0 < alpha <= 2}.
@end deftypefun

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

@page
@node The Levy skew alpha-Stable Distribution
@section The Levy skew alpha-Stable Distribution

@deftypefun double gsl_ran_levy_skew (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}, double @var{beta})
@cindex Levy distribution, skew
@cindex Skew Levy distribution
This function returns a random variate from the Levy skew stable
distribution with scale @var{c}, exponent @var{alpha} and skewness
parameter @var{beta}.  The skewness parameter must lie in the range
@math{[-1,1]}.  The Levy skew stable probability distribution is defined
by a fourier transform,
@tex
\beforedisplay
$$
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha (1-i \beta \sign(t) \tan(\pi\alpha/2)))
$$
\afterdisplay
@end tex
@ifinfo

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

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

The algorithm only works for @c{$0 < \alpha \le 2$}
@math{0 < alpha <= 2}.
@end deftypefun

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

@comment PDF not available because there is no analytic expression for it
@comment
@comment @deftypefun double gsl_ran_levy_pdf (double @var{x}, double @var{mu})
@comment This function computes the probability density @math{p(x)} at @var{x}
@comment for a symmetric Levy distribution with scale parameter @var{mu} and
@comment exponent @var{a}, using the formula given above.
@comment @end deftypefun

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

@page
@node The Gamma Distribution
@section The Gamma Distribution
@deftypefun double gsl_ran_gamma (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex Gamma distribution
This function returns a random variate from the gamma
distribution.  The distribution function is,
@tex
\beforedisplay
$$
p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent