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revision 1.1.1.3.8.2 by bjg, Tue Jul 22 09:23:04 2003 UTC revision 1.1.1.3.8.3 by bjg, Tue Jul 22 09:43:07 2003 UTC
# Line 109  P(x) = \int_@{x@}^@{-\infty@} dx' p(x') Line 109  P(x) = \int_@{x@}^@{-\infty@} dx' p(x')
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{$x=Q^{-1}(Q)$}
# Line 121  They can be used to find confidence limi Line 123  They can be used to find confidence limi
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,
# Line 192  distribution. Line 194  distribution.
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
# Line 257  deviation of one, @var{sigma} = 1. Line 259  deviation of one, @var{sigma} = 1.
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.
# Line 298  deviations @var{sigma_x}, @var{sigma_y} Line 302  deviations @var{sigma_x}, @var{sigma_y}
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    
# Line 344  distribution with mean @var{mu}. Line 348  distribution with mean @var{mu}.
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    
# Line 391  distribution with width @var{a}. Line 395  distribution with width @var{a}.
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    
# Line 430  and exponent @var{b}, using the formula Line 434  and exponent @var{b}, using the formula
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,
# Line 478  distribution with scale parameter @var{a Line 482  distribution with scale parameter @var{a
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    
# Line 522  distribution with scale parameter @var{s Line 526  distribution with scale parameter @var{s
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,
# Line 559  lower limit @var{a}, using the formula g Line 563  lower limit @var{a}, using the formula g
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,
# Line 608  above. Line 612  above.
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,
# Line 647  The algorithm only works for @c{$0 < \al Line 651  The algorithm only works for @c{$0 < \al
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
# Line 705  distributed as an alpha-stable variate, Line 710  distributed as an alpha-stable variate,
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    
# Line 752  distribution with parameters @var{a} and Line 757  distribution with parameters @var{a} and
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    
# Line 798  from @var{a} to @var{b}. Line 804  from @var{a} to @var{b}.
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    
# Line 835  using the formula given above. Line 841  using the formula given above.
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    
# Line 862  X_i = \sum_i Y_i^2 Line 868  X_i = \sum_i Y_i^2
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    
# Line 929  X = @{ (Y_1 / \nu_1) \over (Y_2 / \nu_2) Line 935  X = @{ (Y_1 / \nu_1) \over (Y_2 / \nu_2)
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
# Line 1000  X = @{ Y_1 \over \sqrt@{Y_2 / \nu@} @} Line 1006  X = @{ Y_1 \over \sqrt@{Y_2 / \nu@} @}
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    
# Line 1046  with @var{nu} degrees of freedom. Line 1053  with @var{nu} degrees of freedom.
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    
# Line 1090  parameters @var{a} and @var{b}. Line 1097  parameters @var{a} and @var{b}.
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    
# Line 1134  distribution with scale parameter @var{a Line 1141  distribution with scale parameter @var{a
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    
# Line 1244  Mathematics for the Engineer (1956). Line 1251  Mathematics for the Engineer (1956).
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    
# Line 1291  distribution with scale @var{a} and expo Line 1298  distribution with scale @var{a} and expo
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,
# Line 1338  distribution with parameters @var{a} and Line 1345  distribution with parameters @var{a} and
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    

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