randist.texi

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

for @math{x > 0}.
@comment If @xmath{X} and @xmath{Y} are independent gamma-distributed random
@comment variables of order @xmath{a} and @xmath{b}, then @xmath{X+Y} has a gamma
@comment distribution of order @xmath{a+b}.

@cindex Erlang distribution
The gamma distribution with an integer parameter @var{a} is known as the Erlang distribution.
The variates are computed using the algorithms from Knuth (vol 2).
@end deftypefun

@deftypefun double gsl_ran_gamma_mt (const gsl_rng * @var{r}, double @var{a}, double @var{b})
This function returns a gamma variate using the Marsaglia-Tsang fast gamma method.
@end deftypefun

@deftypefun double gsl_ran_gamma_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for a gamma distribution with parameters @var{a} and @var{b}, using the
formula given above.
@end deftypefun

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

@deftypefun double gsl_cdf_gamma_P (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_gamma_Q (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_gamma_Pinv (double @var{P}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_gamma_Qinv (double @var{Q}, double @var{a}, double @var{b})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the gamma
distribution with parameters @var{a} and @var{b}.
@end deftypefun

@page
@node The Flat (Uniform) Distribution
@section The Flat (Uniform) Distribution
@deftypefun double gsl_ran_flat (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex flat distribution
@cindex uniform distribution
This function returns a random variate from the flat (uniform)
distribution from @var{a} to @var{b}. The distribution is,
@tex
\beforedisplay
$$
p(x) dx = {1 \over (b-a)} dx
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
if @c{$a \le x < b$}
@math{a <= x < b} and 0 otherwise.
@end deftypefun

@deftypefun double gsl_ran_flat_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for a uniform distribution from @var{a} to @var{b}, using the formula
given above.
@end deftypefun

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

@deftypefun double gsl_cdf_flat_P (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_flat_Q (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_flat_Pinv (double @var{P}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_flat_Qinv (double @var{Q}, double @var{a}, double @var{b})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for a uniform distribution
from @var{a} to @var{b}.
@end deftypefun


@page
@node The Lognormal Distribution
@section The Lognormal Distribution
@deftypefun double gsl_ran_lognormal (const gsl_rng * @var{r}, double @var{zeta}, double @var{sigma})
@cindex Lognormal distribution
This function returns a random variate from the lognormal
distribution.  The distribution function is,
@tex
\beforedisplay
$$
p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
for @math{x > 0}.
@end deftypefun

@deftypefun double gsl_ran_lognormal_pdf (double @var{x}, double @var{zeta}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a lognormal distribution with parameters @var{zeta} and @var{sigma},
using the formula given above.
@end deftypefun

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

@deftypefun double gsl_cdf_lognormal_P (double @var{x}, double @var{zeta}, double @var{sigma})
@deftypefunx double gsl_cdf_lognormal_Q (double @var{x}, double @var{zeta}, double @var{sigma})
@deftypefunx double gsl_cdf_lognormal_Pinv (double @var{P}, double @var{zeta}, double @var{sigma})
@deftypefunx double gsl_cdf_lognormal_Qinv (double @var{Q}, double @var{zeta}, double @var{sigma})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the lognormal
distribution with parameters @var{zeta} and @var{sigma}.
@end deftypefun


@page
@node The Chi-squared Distribution
@section The Chi-squared Distribution
The chi-squared distribution arises in statistics.  If @math{Y_i} are
@math{n} independent gaussian random variates with unit variance then the
sum-of-squares,
@tex
\beforedisplay
$$
X_i = \sum_i Y_i^2
$$
\afterdisplay
@end tex
@ifinfo

@example
X_i = \sum_i Y_i^2
@end example

@end ifinfo
@noindent
has a chi-squared distribution with @math{n} degrees of freedom.

@deftypefun double gsl_ran_chisq (const gsl_rng * @var{r}, double @var{nu})
@cindex Chi-squared distribution
This function returns a random variate from the chi-squared distribution
with @var{nu} degrees of freedom. The distribution function is,
@tex
\beforedisplay
$$
p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
$$
\afterdisplay
@end tex
@ifinfo

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

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

@deftypefun double gsl_ran_chisq_pdf (double @var{x}, double @var{nu})
This function computes the probability density @math{p(x)} at @var{x}
for a chi-squared distribution with @var{nu} degrees of freedom, using
the formula given above.
@end deftypefun

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

@deftypefun double gsl_cdf_chisq_P (double @var{x}, double @var{nu})
@deftypefunx double gsl_cdf_chisq_Q (double @var{x}, double @var{nu})
@deftypefunx double gsl_cdf_chisq_Pinv (double @var{P}, double @var{nu})
@deftypefunx double gsl_cdf_chisq_Qinv (double @var{Q}, double @var{nu})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the chi-squared
distribution with @var{nu} degrees of freedom.
@end deftypefun



@page
@node The F-distribution
@section The F-distribution
The F-distribution arises in statistics.  If @math{Y_1} and @math{Y_2}
are chi-squared deviates with @math{\nu_1} and @math{\nu_2} degrees of
freedom then the ratio,
@tex
\beforedisplay
$$
X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
has an F-distribution @math{F(x;\nu_1,\nu_2)}.

@deftypefun double gsl_ran_fdist (const gsl_rng * @var{r}, double @var{nu1}, double @var{nu2})
@cindex F-distribution
This function returns a random variate from the F-distribution with degrees of freedom @var{nu1} and @var{nu2}. The distribution function is,
@tex
\beforedisplay
$$
p(x) dx = 
   { \Gamma((\nu_1 + \nu_2)/2)
        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 
   \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 
   x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
$$
\afterdisplay
@end tex
@ifinfo

@example
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@}
@end example

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

@deftypefun double gsl_ran_fdist_pdf (double @var{x}, double @var{nu1}, double @var{nu2})
This function computes the probability density @math{p(x)} at @var{x}
for an F-distribution with @var{nu1} and @var{nu2} degrees of freedom,
using the formula given above.
@end deftypefun

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

@deftypefun double gsl_cdf_fdist_P (double @var{x}, double @var{nu1}, double @var{nu2})
@deftypefunx double gsl_cdf_fdist_Q (double @var{x}, double @var{nu1}, double @var{nu2})
@deftypefunx double gsl_cdf_fdist_Pinv (double @var{P}, double @var{nu1}, double @var{nu2})
@deftypefunx double gsl_cdf_fdist_Qinv (double @var{Q}, double @var{nu1}, double @var{nu2})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the F-distribution
with @var{nu1} and @var{nu2} degrees of freedom.
@end deftypefun

@page
@node The t-distribution
@section The t-distribution
The t-distribution arises in statistics.  If @math{Y_1} has a normal
distribution and @math{Y_2} has a chi-squared distribution with
@math{\nu} degrees of freedom then the ratio,
@tex
\beforedisplay
$$
X = { Y_1 \over \sqrt{Y_2 / \nu} }
$$
\afterdisplay
@end tex
@ifinfo

@example
X = @{ Y_1 \over \sqrt@{Y_2 / \nu@} @}
@end example

@end ifinfo
@noindent
has a t-distribution @math{t(x;\nu)} with @math{\nu} degrees of freedom.

@deftypefun double gsl_ran_tdist (const gsl_rng * @var{r}, double @var{nu})
@cindex t-distribution
@cindex Student t-distribution
This function returns a random variate from the t-distribution.  The
distribution function is,
@tex
\beforedisplay
$$
p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
   (1 + x^2/\nu)^{-(\nu + 1)/2} dx
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
for @math{-\infty < x < +\infty}.
@end deftypefun

@deftypefun double gsl_ran_tdist_pdf (double @var{x}, double @var{nu})
This function computes the probability density @math{p(x)} at @var{x}
for a t-distribution with @var{nu} degrees of freedom, using the formula
given above.
@end deftypefun

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

@deftypefun double gsl_cdf_tdist_P (double @var{x}, double @var{nu})
@deftypefunx double gsl_cdf_tdist_Q (double @var{x}, double @var{nu})
@deftypefunx double gsl_cdf_tdist_Pinv (double @var{P}, double @var{nu})
@deftypefunx double gsl_cdf_tdist_Qinv (double @var{Q}, double @var{nu})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the t-distribution
with @var{nu} degrees of freedom.
@end deftypefun

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

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

@end ifinfo
@noindent
for @c{$0 \le x \le 1$}
@math{0 <= x <= 1}.
@end deftypefun

@deftypefun double gsl_ran_beta_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for a beta distribution with parameters @var{a} and @var{b}, using the
formula given above.
@end deftypefun

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

@deftypefun double gsl_cdf_beta_P (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_beta_Q (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_beta_Pinv (double @var{P}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_beta_Qinv (double @var{Q}, double @var{a}, double @var{b})
These functions compute the cumulative distribution functions
@math{P(x)}, @math{Q(x)} and their inverses for the beta
distribution with parameters @var{a} and @var{b}.
@end deftypefun

@page
@node The Logistic Distribution
@section The Logistic Distribution

@deftypefun double gsl_ran_logistic (const gsl_rng * @var{r}, double @var{a})
@cindex Logistic distribution
This function returns a random variate from the logistic
distribution.  The distribution function is,
@tex
\beforedisplay
$$
p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo