randist.texi

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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 deftypefn

@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.

@deftypefn Random 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 deftypefn

@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)}.

@deftypefn Random 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 deftypefn

@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})
These functions compute the cumulative distribution functions
@math{P(x)} and @math{Q(x)} 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.

@deftypefn Random 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 deftypefn

@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
@deftypefn Random 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 deftypefn

@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})
These functions compute the cumulative distribution functions
@math{P(x)} and @math{Q(x)} for the beta distribution with
parameters @var{a} and @var{b}.
@end deftypefun

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

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

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

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

@deftypefun double gsl_cdf_logistic_P (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_logistic_Q (double @var{x}, double @var{a})
@deftypefunx double gsl_cdf_logistic_Pinv (double @var{P}, double @var{a})
@deftypefunx double gsl_cdf_logistic_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 logistic
distribution with scale parameter @var{a}.
@end deftypefun

@page
@node The Pareto Distribution
@section The Pareto Distribution
@deftypefn Random double gsl_ran_pareto (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex Pareto distribution
This function returns a random variate from the Pareto distribution of
order @var{a}.  The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = (a/b) / (x/b)^{a+1} dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = (a/b) / (x/b)^@{a+1@} dx
@end example
@end ifinfo
@noindent
for @c{$x \ge b$}
@math{x >= b}.
@end deftypefn

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

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

@deftypefun double gsl_cdf_pareto_P (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_pareto_Q (double @var{x}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_pareto_Pinv (double @var{P}, double @var{a}, double @var{b})
@deftypefunx double gsl_cdf_pareto_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 Pareto
distribution with exponent @var{a} and scale @var{b}.
@end deftypefun

@page
@node The Spherical Distribution (2D & 3D)
@section The Spherical Distribution (2D & 3D)

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.

@deftypefn Random void gsl_ran_dir_2d (const gsl_rng * @var{r}, double *@var{x}, double *@var{y})
@deftypefnx Random void gsl_ran_dir_2d_trig_method (const gsl_rng * @var{r}, double *@var{x}, double *@var{y})
@cindex 2D random direction vector
@cindex direction vector, random 2D
@cindex spherical random variates, 2D
This function returns a random direction vector @math{v} =
(@var{x},@var{y}) in two dimensions.  The vector is normalized such that
@math{|v|^2 = x^2 + y^2 = 1}.  The obvious way to do this is to take a
uniform random number between 0 and @math{2\pi} and let @var{x} and
@var{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 @var{x} and
@var{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 @c{$\sqrt{x^2 + y^2}$}
@math{\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, @var{u} and @var{v} are chosen at random from the
interior of a unit circle, and then @math{x=(u^2-v^2)/(u^2+v^2)} and
@math{y=uv/(u^2+v^2)}.
@end deftypefn



@deftypefn Random void gsl_ran_dir_3d (const gsl_rng * @var{r}, double *@var{x}, double *@var{y}, double * @var{z})
@cindex 3D random direction vector