109 |
@noindent |
@noindent |
110 |
and gives the probability of a variate taking a greater than @math{x}. |
and gives the probability of a variate taking a greater than @math{x}. |
111 |
The upper and lower cumulative distribution functions are related by |
The upper and lower cumulative distribution functions are related by |
112 |
@math{P(x) + Q(x) = 1} |
@math{P(x) + Q(x) = 1} and satisfy @c{$0 \le P(x) \le 1$} |
113 |
|
@math{0 <= P(x) <= 1}, @c{$0 \le Q(x) \le 1$} |
114 |
|
@math{0 <= Q(x) <= 1}. |
115 |
|
|
116 |
The inverse cumulative distributions, @c{$x=P^{-1}(P)$} |
The inverse cumulative distributions, @c{$x=P^{-1}(P)$} |
117 |
@math{x=P^@{-1@}(P)} and @c{$x=Q^{-1}(Q)$} |
@math{x=P^@{-1@}(P)} and @c{$x=Q^{-1}(Q)$} |
123 |
@node The Gaussian Distribution |
@node The Gaussian Distribution |
124 |
@section The Gaussian Distribution |
@section The Gaussian Distribution |
125 |
@deftypefn Random double gsl_ran_gaussian (const gsl_rng * @var{r}, double @var{sigma}) |
@deftypefn Random double gsl_ran_gaussian (const gsl_rng * @var{r}, double @var{sigma}) |
126 |
@cindex Gaussian random variates |
@cindex Gaussian distribution |
127 |
This function returns a Gaussian random variate, with mean zero and |
This function returns a Gaussian random variate, with mean zero and |
128 |
standard deviation @var{sigma}. The probability distribution for |
standard deviation @var{sigma}. The probability distribution for |
129 |
Gaussian random variates is, |
Gaussian random variates is, |
194 |
@node The Gaussian Tail Distribution |
@node The Gaussian Tail Distribution |
195 |
@section The Gaussian Tail Distribution |
@section The Gaussian Tail Distribution |
196 |
@deftypefn Random double gsl_ran_gaussian_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma}) |
@deftypefn Random double gsl_ran_gaussian_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma}) |
197 |
@cindex Gaussian Tail random variates |
@cindex Gaussian Tail distribution |
198 |
This function provides random variates from the upper tail of a Gaussian |
This function provides random variates from the upper tail of a Gaussian |
199 |
distribution with standard deviation @var{sigma}. The values returned |
distribution with standard deviation @var{sigma}. The values returned |
200 |
are larger than the lower limit @var{a}, which must be positive. The |
are larger than the lower limit @var{a}, which must be positive. The |
259 |
@section The Bivariate Gaussian Distribution |
@section The Bivariate Gaussian Distribution |
260 |
|
|
261 |
@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}) |
@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}) |
262 |
@cindex Gaussian random variates |
@cindex Bivariate Gaussian distribution |
263 |
|
@cindex Two-dimensional Gaussian distribution |
264 |
|
@cindex Gaussian distribution, bivariate |
265 |
This function generates a pair of correlated gaussian variates, with |
This function generates a pair of correlated gaussian variates, with |
266 |
mean zero, correlation coefficient @var{rho} and standard deviations |
mean zero, correlation coefficient @var{rho} and standard deviations |
267 |
@var{sigma_x} and @var{sigma_y} in the @math{x} and @math{y} directions. |
@var{sigma_x} and @var{sigma_y} in the @math{x} and @math{y} directions. |
302 |
@node The Exponential Distribution |
@node The Exponential Distribution |
303 |
@section The Exponential Distribution |
@section The Exponential Distribution |
304 |
@deftypefn Random double gsl_ran_exponential (const gsl_rng * @var{r}, double @var{mu}) |
@deftypefn Random double gsl_ran_exponential (const gsl_rng * @var{r}, double @var{mu}) |
305 |
@cindex Exponential random variates |
@cindex Exponential distribution |
306 |
This function returns a random variate from the exponential distribution |
This function returns a random variate from the exponential distribution |
307 |
with mean @var{mu}. The distribution is, |
with mean @var{mu}. The distribution is, |
308 |
|
|
348 |
@node The Laplace Distribution |
@node The Laplace Distribution |
349 |
@section The Laplace Distribution |
@section The Laplace Distribution |
350 |
@deftypefn Random double gsl_ran_laplace (const gsl_rng * @var{r}, double @var{a}) |
@deftypefn Random double gsl_ran_laplace (const gsl_rng * @var{r}, double @var{a}) |
351 |
@cindex two-sided exponential random variates |
@cindex Two-sided exponential distribution |
352 |
@cindex Laplace distribution random variates |
@cindex Laplace distribution |
353 |
This function returns a random variate from the the Laplace distribution |
This function returns a random variate from the the Laplace distribution |
354 |
with width @var{a}. The distribution is, |
with width @var{a}. The distribution is, |
355 |
|
|
395 |
@node The Exponential Power Distribution |
@node The Exponential Power Distribution |
396 |
@section The Exponential Power Distribution |
@section The Exponential Power Distribution |
397 |
@deftypefn Random double gsl_ran_exppow (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_exppow (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
398 |
@cindex Exponential power distribution, random variates |
@cindex Exponential power distribution |
399 |
This function returns a random variate from the exponential power distribution |
This function returns a random variate from the exponential power distribution |
400 |
with scale parameter @var{a} and exponent @var{b}. The distribution is, |
with scale parameter @var{a} and exponent @var{b}. The distribution is, |
401 |
|
|
434 |
@node The Cauchy Distribution |
@node The Cauchy Distribution |
435 |
@section The Cauchy Distribution |
@section The Cauchy Distribution |
436 |
@deftypefn Random double gsl_ran_cauchy (const gsl_rng * @var{r}, double @var{a}) |
@deftypefn Random double gsl_ran_cauchy (const gsl_rng * @var{r}, double @var{a}) |
437 |
@cindex Cauchy random variates |
@cindex Cauchy distribution |
438 |
This function returns a random variate from the Cauchy distribution with |
This function returns a random variate from the Cauchy distribution with |
439 |
scale parameter @var{a}. The probability distribution for Cauchy |
scale parameter @var{a}. The probability distribution for Cauchy |
440 |
random variates is, |
random variates is, |
482 |
@node The Rayleigh Distribution |
@node The Rayleigh Distribution |
483 |
@section The Rayleigh Distribution |
@section The Rayleigh Distribution |
484 |
@deftypefn Random double gsl_ran_rayleigh (const gsl_rng * @var{r}, double @var{sigma}) |
@deftypefn Random double gsl_ran_rayleigh (const gsl_rng * @var{r}, double @var{sigma}) |
485 |
@cindex Rayleigh random variates |
@cindex Rayleigh distribution |
486 |
This function returns a random variate from the Rayleigh distribution with |
This function returns a random variate from the Rayleigh distribution with |
487 |
scale parameter @var{sigma}. The distribution is, |
scale parameter @var{sigma}. The distribution is, |
488 |
|
|
526 |
@node The Rayleigh Tail Distribution |
@node The Rayleigh Tail Distribution |
527 |
@section The Rayleigh Tail Distribution |
@section The Rayleigh Tail Distribution |
528 |
@deftypefn Random double gsl_ran_rayleigh_tail (const gsl_rng * @var{r}, double @var{a} double @var{sigma}) |
@deftypefn Random double gsl_ran_rayleigh_tail (const gsl_rng * @var{r}, double @var{a} double @var{sigma}) |
529 |
@cindex Rayleigh Tail random variates |
@cindex Rayleigh Tail distribution |
530 |
This function returns a random variate from the tail of the Rayleigh |
This function returns a random variate from the tail of the Rayleigh |
531 |
distribution with scale parameter @var{sigma} and a lower limit of |
distribution with scale parameter @var{sigma} and a lower limit of |
532 |
@var{a}. The distribution is, |
@var{a}. The distribution is, |
563 |
@node The Landau Distribution |
@node The Landau Distribution |
564 |
@section The Landau Distribution |
@section The Landau Distribution |
565 |
@deftypefn Random double gsl_ran_landau (const gsl_rng * @var{r}) |
@deftypefn Random double gsl_ran_landau (const gsl_rng * @var{r}) |
566 |
@cindex Landau random variates |
@cindex Landau distribution |
567 |
This function returns a random variate from the Landau distribution. The |
This function returns a random variate from the Landau distribution. The |
568 |
probability distribution for Landau random variates is defined |
probability distribution for Landau random variates is defined |
569 |
analytically by the complex integral, |
analytically by the complex integral, |
612 |
@node The Levy alpha-Stable Distributions |
@node The Levy alpha-Stable Distributions |
613 |
@section The Levy alpha-Stable Distributions |
@section The Levy alpha-Stable Distributions |
614 |
@deftypefn Random double gsl_ran_levy (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}) |
@deftypefn Random double gsl_ran_levy (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}) |
615 |
@cindex Levy distribution, random variates |
@cindex Levy distribution |
616 |
This function returns a random variate from the Levy symmetric stable |
This function returns a random variate from the Levy symmetric stable |
617 |
distribution with scale @var{c} and exponent @var{alpha}. The symmetric |
distribution with scale @var{c} and exponent @var{alpha}. The symmetric |
618 |
stable probability distribution is defined by a fourier transform, |
stable probability distribution is defined by a fourier transform, |
651 |
@section The Levy skew alpha-Stable Distribution |
@section The Levy skew alpha-Stable Distribution |
652 |
|
|
653 |
@deftypefn Random double gsl_ran_levy_skew (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}, double @var{beta}) |
@deftypefn Random double gsl_ran_levy_skew (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}, double @var{beta}) |
654 |
@cindex Levy distribution, random variates |
@cindex Levy distribution, skew |
655 |
|
@cindex Skew Levy distribution |
656 |
This function returns a random variate from the Levy skew stable |
This function returns a random variate from the Levy skew stable |
657 |
distribution with scale @var{c}, exponent @var{alpha} and skewness |
distribution with scale @var{c}, exponent @var{alpha} and skewness |
658 |
parameter @var{beta}. The skewness parameter must lie in the range |
parameter @var{beta}. The skewness parameter must lie in the range |
710 |
@node The Gamma Distribution |
@node The Gamma Distribution |
711 |
@section The Gamma Distribution |
@section The Gamma Distribution |
712 |
@deftypefn Random double gsl_ran_gamma (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_gamma (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
713 |
@cindex Gamma distribution random variates |
@cindex Gamma distribution |
714 |
This function returns a random variate from the gamma |
This function returns a random variate from the gamma |
715 |
distribution. The distribution function is, |
distribution. The distribution function is, |
716 |
|
|
757 |
@node The Flat (Uniform) Distribution |
@node The Flat (Uniform) Distribution |
758 |
@section The Flat (Uniform) Distribution |
@section The Flat (Uniform) Distribution |
759 |
@deftypefn Random double gsl_ran_flat (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_flat (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
760 |
@cindex flat distribution random variates |
@cindex flat distribution |
761 |
|
@cindex uniform distribution |
762 |
This function returns a random variate from the flat (uniform) |
This function returns a random variate from the flat (uniform) |
763 |
distribution from @var{a} to @var{b}. The distribution is, |
distribution from @var{a} to @var{b}. The distribution is, |
764 |
|
|
804 |
@node The Lognormal Distribution |
@node The Lognormal Distribution |
805 |
@section The Lognormal Distribution |
@section The Lognormal Distribution |
806 |
@deftypefn Random double gsl_ran_lognormal (const gsl_rng * @var{r}, double @var{zeta}, double @var{sigma}) |
@deftypefn Random double gsl_ran_lognormal (const gsl_rng * @var{r}, double @var{zeta}, double @var{sigma}) |
807 |
@cindex Lognormal random variates |
@cindex Lognormal distribution |
808 |
This function returns a random variate from the lognormal |
This function returns a random variate from the lognormal |
809 |
distribution. The distribution function is, |
distribution. The distribution function is, |
810 |
|
|
841 |
@deftypefunx double gsl_cdf_lognormal_Qinv (double @var{Q}, double @var{zeta}, double @var{sigma}) |
@deftypefunx double gsl_cdf_lognormal_Qinv (double @var{Q}, double @var{zeta}, double @var{sigma}) |
842 |
These functions compute the cumulative distribution functions |
These functions compute the cumulative distribution functions |
843 |
@math{P(x)}, @math{Q(x)} and their inverses for the lognormal |
@math{P(x)}, @math{Q(x)} and their inverses for the lognormal |
844 |
distribution with parameters @var{zeta} and @var{sigma}, |
distribution with parameters @var{zeta} and @var{sigma}. |
845 |
@end deftypefun |
@end deftypefun |
846 |
|
|
847 |
|
|
868 |
has a chi-squared distribution with @math{n} degrees of freedom. |
has a chi-squared distribution with @math{n} degrees of freedom. |
869 |
|
|
870 |
@deftypefn Random double gsl_ran_chisq (const gsl_rng * @var{r}, double @var{nu}) |
@deftypefn Random double gsl_ran_chisq (const gsl_rng * @var{r}, double @var{nu}) |
871 |
@cindex Chi-squared random variates |
@cindex Chi-squared distribution |
872 |
This function returns a random variate from the chi-squared distribution |
This function returns a random variate from the chi-squared distribution |
873 |
with @var{nu} degrees of freedom. The distribution function is, |
with @var{nu} degrees of freedom. The distribution function is, |
874 |
|
|
935 |
has an F-distribution @math{F(x;\nu_1,\nu_2)}. |
has an F-distribution @math{F(x;\nu_1,\nu_2)}. |
936 |
|
|
937 |
@deftypefn Random double gsl_ran_fdist (const gsl_rng * @var{r}, double @var{nu1}, double @var{nu2}) |
@deftypefn Random double gsl_ran_fdist (const gsl_rng * @var{r}, double @var{nu1}, double @var{nu2}) |
938 |
@cindex F-distribution random variates |
@cindex F-distribution |
939 |
This function returns a random variate from the F-distribution with degrees of freedom @var{nu1} and @var{nu2}. The distribution function is, |
This function returns a random variate from the F-distribution with degrees of freedom @var{nu1} and @var{nu2}. The distribution function is, |
940 |
|
|
941 |
@tex |
@tex |
1006 |
has a t-distribution @math{t(x;\nu)} with @math{\nu} degrees of freedom. |
has a t-distribution @math{t(x;\nu)} with @math{\nu} degrees of freedom. |
1007 |
|
|
1008 |
@deftypefn Random double gsl_ran_tdist (const gsl_rng * @var{r}, double @var{nu}) |
@deftypefn Random double gsl_ran_tdist (const gsl_rng * @var{r}, double @var{nu}) |
1009 |
@cindex t-distribution random variates |
@cindex t-distribution |
1010 |
|
@cindex Student t-distribution |
1011 |
This function returns a random variate from the t-distribution. The |
This function returns a random variate from the t-distribution. The |
1012 |
distribution function is, |
distribution function is, |
1013 |
|
|
1053 |
@node The Beta Distribution |
@node The Beta Distribution |
1054 |
@section The Beta Distribution |
@section The Beta Distribution |
1055 |
@deftypefn Random double gsl_ran_beta (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_beta (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
1056 |
@cindex Beta distribution random variates |
@cindex Beta distribution |
1057 |
This function returns a random variate from the beta |
This function returns a random variate from the beta |
1058 |
distribution. The distribution function is, |
distribution. The distribution function is, |
1059 |
|
|
1097 |
@section The Logistic Distribution |
@section The Logistic Distribution |
1098 |
|
|
1099 |
@deftypefn Random double gsl_ran_logistic (const gsl_rng * @var{r}, double @var{a}) |
@deftypefn Random double gsl_ran_logistic (const gsl_rng * @var{r}, double @var{a}) |
1100 |
@cindex Logistic random variates |
@cindex Logistic distribution |
1101 |
This function returns a random variate from the logistic |
This function returns a random variate from the logistic |
1102 |
distribution. The distribution function is, |
distribution. The distribution function is, |
1103 |
|
|
1141 |
@node The Pareto Distribution |
@node The Pareto Distribution |
1142 |
@section The Pareto Distribution |
@section The Pareto Distribution |
1143 |
@deftypefn Random double gsl_ran_pareto (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_pareto (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
1144 |
@cindex Pareto random variates |
@cindex Pareto distribution |
1145 |
This function returns a random variate from the Pareto distribution of |
This function returns a random variate from the Pareto distribution of |
1146 |
order @var{a}. The distribution function is, |
order @var{a}. The distribution function is, |
1147 |
|
|
1251 |
@node The Weibull Distribution |
@node The Weibull Distribution |
1252 |
@section The Weibull Distribution |
@section The Weibull Distribution |
1253 |
@deftypefn Random double gsl_ran_weibull (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_weibull (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
1254 |
@cindex Weibull distribution random variates |
@cindex Weibull distribution |
1255 |
This function returns a random variate from the Weibull distribution. The |
This function returns a random variate from the Weibull distribution. The |
1256 |
distribution function is, |
distribution function is, |
1257 |
|
|
1298 |
@node The Type-1 Gumbel Distribution |
@node The Type-1 Gumbel Distribution |
1299 |
@section The Type-1 Gumbel Distribution |
@section The Type-1 Gumbel Distribution |
1300 |
@deftypefn Random double gsl_ran_gumbel1 (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_gumbel1 (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
1301 |
@cindex Gumbel distribution (Type 1), random variates |
@cindex Gumbel distribution (Type 1) |
1302 |
@cindex Type 1 Gumbel distribution, random variates |
@cindex Type 1 Gumbel distribution, random variates |
1303 |
This function returns a random variate from the Type-1 Gumbel |
This function returns a random variate from the Type-1 Gumbel |
1304 |
distribution. The Type-1 Gumbel distribution function is, |
distribution. The Type-1 Gumbel distribution function is, |
1345 |
@node The Type-2 Gumbel Distribution |
@node The Type-2 Gumbel Distribution |
1346 |
@section The Type-2 Gumbel Distribution |
@section The Type-2 Gumbel Distribution |
1347 |
@deftypefn Random double gsl_ran_gumbel2 (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
@deftypefn Random double gsl_ran_gumbel2 (const gsl_rng * @var{r}, double @var{a}, double @var{b}) |
1348 |
@cindex Gumbel distribution (Type 2), random variates |
@cindex Gumbel distribution (Type 2) |
1349 |
@cindex Type 2 Gumbel distribution, random variate |
@cindex Type 2 Gumbel distribution |
1350 |
This function returns a random variate from the Type-2 Gumbel |
This function returns a random variate from the Type-2 Gumbel |
1351 |
distribution. The Type-2 Gumbel distribution function is, |
distribution. The Type-2 Gumbel distribution function is, |
1352 |
|
|