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revision 1.1.1.3.8.1 by bjg, Sun Jul 20 15:02:50 2003 UTC revision 1.1.1.3.8.2 by bjg, Tue Jul 22 09:23:04 2003 UTC
# Line 1  Line 1 
1  @cindex random number distributions  @cindex random number distributions
2    @cindex cumulative distribution functions (CDFs)
3    @cindex CDFs, cumulative distribution functions
4    @cindex inverse cumulative distribution functions
5    @cindex quantile functions
6  This chapter describes functions for generating random variates and  This chapter describes functions for generating random variates and
7  computing their probability distributions.  Samples from the  computing their probability distributions.  Samples from the
8  distributions described in this chapter can be obtained using any of the  distributions described in this chapter can be obtained using any of the
# Line 108  and gives the probability of a variate t Line 111  and gives the probability of a variate t
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}
113    
114  The inverse cumulative distributions, @c{$P^{-1}(x)$}  The inverse cumulative distributions, @c{$x=P^{-1}(P)$}
115  @math{P^@{-1@}(x)} and @c{$Q^{-1}(x)$}  @math{x=P^@{-1@}(P)} and @c{$x=Q^{-1}(Q)$}
116  @math{Q^@{-1@}(x)} give the value of @math{x}  @math{x=Q^@{-1@}(Q)} give the values of @math{x}
117  which corresponds to a given value of @math{P} or {Q}.    which correspond to a specific value of @math{P} or {Q}.  
118  They can be used to find confidence limits from probability values.  They can be used to find confidence limits from probability values.
119    
120  @page  @page
# Line 172  are equivalent to the functions above wi Line 175  are equivalent to the functions above wi
175  @deftypefunx double gsl_cdf_gaussian_Pinv (double @var{P}, double @var{sigma})  @deftypefunx double gsl_cdf_gaussian_Pinv (double @var{P}, double @var{sigma})
176  @deftypefunx double gsl_cdf_gaussian_Qinv (double @var{Q}, double @var{sigma})  @deftypefunx double gsl_cdf_gaussian_Qinv (double @var{Q}, double @var{sigma})
177  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
178  @math{P(x)} and @math{Q(x)}, and their inverses, for the Gaussian  @math{P(x)}, @math{Q(x)} and their inverses for the Gaussian
179  distribution with standard deviation @var{sigma}.  distribution with standard deviation @var{sigma}.
180  @end deftypefun  @end deftypefun
181    
# Line 181  distribution with standard deviation @va Line 184  distribution with standard deviation @va
184  @deftypefunx double gsl_cdf_ugaussian_Pinv (double @var{P})  @deftypefunx double gsl_cdf_ugaussian_Pinv (double @var{P})
185  @deftypefunx double gsl_cdf_ugaussian_Qinv (double @var{Q})  @deftypefunx double gsl_cdf_ugaussian_Qinv (double @var{Q})
186  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
187  @math{P(x)} and @math{Q(x)}, and their inverses, for the unit Gaussian  @math{P(x)}, @math{Q(x)} and their inverses for the unit Gaussian
188  distribution.  distribution.
189  @end deftypefun  @end deftypefun
190    
# Line 333  given above. Line 336  given above.
336  @deftypefunx double gsl_cdf_exponential_Pinv (double @var{P}, double @var{mu})  @deftypefunx double gsl_cdf_exponential_Pinv (double @var{P}, double @var{mu})
337  @deftypefunx double gsl_cdf_exponential_Qinv (double @var{Q}, double @var{mu})  @deftypefunx double gsl_cdf_exponential_Qinv (double @var{Q}, double @var{mu})
338  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
339  @math{P(x)} and @math{Q(x)}, and their inverses, for the exponential  @math{P(x)}, @math{Q(x)} and their inverses for the exponential
340  distribution with mean @var{mu}.  distribution with mean @var{mu}.
341  @end deftypefun  @end deftypefun
342    
# Line 365  for @math{-\infty < x < \infty}. Line 368  for @math{-\infty < x < \infty}.
368    
369  @deftypefun double gsl_ran_laplace_pdf (double @var{x}, double @var{a})  @deftypefun double gsl_ran_laplace_pdf (double @var{x}, double @var{a})
370  This function computes the probability density @math{p(x)} at @var{x}  This function computes the probability density @math{p(x)} at @var{x}
371  for a Laplace distribution with mean @var{a}, using the formula  for a Laplace distribution with width @var{a}, using the formula
372  given above.  given above.
373  @end deftypefun  @end deftypefun
374    
# Line 374  given above. Line 377  given above.
377  \centerline{\input rand-laplace.tex}  \centerline{\input rand-laplace.tex}
378  @end tex  @end tex
379    
380    @deftypefun double gsl_cdf_laplace_P (const double @var{x}, const double @var{a})
381    @deftypefunx double gsl_cdf_laplace_Q (const double @var{x}, const double @var{a})
382    @deftypefunx double gsl_cdf_laplace_Pinv (const double @var{P}, const double @var{a})
383    @deftypefunx double gsl_cdf_laplace_Qinv (const double @var{Q}, const double @var{a})
384    These functions compute the cumulative distribution functions
385    @math{P(x)}, @math{Q(x)} and their inverses for the Laplace
386    distribution with width @var{a}.
387    @end deftypefun
388    
389    
390  @page  @page
391  @node The Exponential Power Distribution  @node The Exponential Power Distribution
392  @section The Exponential Power Distribution  @section The Exponential Power Distribution
# Line 451  given above. Line 464  given above.
464  \centerline{\input rand-cauchy.tex}  \centerline{\input rand-cauchy.tex}
465  @end tex  @end tex
466    
467    @deftypefun double gsl_cdf_cauchy_P (const double @var{x}, const double @var{a})
468    @deftypefunx double gsl_cdf_cauchy_Q (const double @var{x}, const double @var{a})
469    @deftypefunx double gsl_cdf_cauchy_Pinv (const double @var{P}, const double @var{a})
470    @deftypefunx double gsl_cdf_cauchy_Qinv (const double @var{Q}, const double @var{a})
471    These functions compute the cumulative distribution functions
472    @math{P(x)}, @math{Q(x)} and their inverses for the Cauchy
473    distribution with scale parameter @var{a}.
474    @end deftypefun
475    
476    
477  @page  @page
478  @node The Rayleigh Distribution  @node The Rayleigh Distribution
479  @section The Rayleigh Distribution  @section The Rayleigh Distribution
# Line 486  formula given above. Line 509  formula given above.
509  \centerline{\input rand-rayleigh.tex}  \centerline{\input rand-rayleigh.tex}
510  @end tex  @end tex
511    
512    @deftypefun double gsl_cdf_rayleigh_P (const double @var{x}, const double @var{sigma})
513    @deftypefunx double gsl_cdf_rayleigh_Q (const double @var{x}, const double @var{sigma})
514    @deftypefunx double gsl_cdf_rayleigh_Pinv (const double @var{P}, const double @var{sigma})
515    @deftypefunx double gsl_cdf_rayleigh_Qinv (const double @var{Q}, const double @var{sigma})
516    These functions compute the cumulative distribution functions
517    @math{P(x)}, @math{Q(x)} and their inverses for the Rayleigh
518    distribution with scale parameter @var{sigma}.
519    @end deftypefun
520    
521  @page  @page
522  @node The Rayleigh Tail Distribution  @node The Rayleigh Tail Distribution
523  @section The Rayleigh Tail Distribution  @section The Rayleigh Tail Distribution
# Line 712  formula given above. Line 744  formula given above.
744  @deftypefunx double gsl_cdf_gamma_Pinv (double @var{P}, double @var{a}, double @var{b})  @deftypefunx double gsl_cdf_gamma_Pinv (double @var{P}, double @var{a}, double @var{b})
745  @deftypefunx double gsl_cdf_gamma_Qinv (double @var{Q}, double @var{a}, double @var{b})  @deftypefunx double gsl_cdf_gamma_Qinv (double @var{Q}, double @var{a}, double @var{b})
746  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
747  @math{P(x)} and @math{Q(x)}, and their inverses, for the gamma  @math{P(x)}, @math{Q(x)} and their inverses for the gamma
748  distribution with parameters @var{a} and @var{b}.  distribution with parameters @var{a} and @var{b}.
749  @end deftypefun  @end deftypefun
750    
# Line 752  given above. Line 784  given above.
784  \centerline{\input rand-flat.tex}  \centerline{\input rand-flat.tex}
785  @end tex  @end tex
786    
787    @deftypefun double gsl_cdf_flat_P (double @var{x}, double @var{a}, double @var{b})
788    @deftypefunx double gsl_cdf_flat_Q (double @var{x}, double @var{a}, double @var{b})
789    @deftypefunx double gsl_cdf_flat_Pinv (double @var{P}, double @var{a}, double @var{b})
790    @deftypefunx double gsl_cdf_flat_Qinv (double @var{Q}, double @var{a}, double @var{b})
791    These functions compute the cumulative distribution functions
792    @math{P(x)}, @math{Q(x)} and their inverses for a uniform distribution
793    from @var{a} to @var{b}.
794    @end deftypefun
795    
796    
797  @page  @page
798  @node The Lognormal Distribution  @node The Lognormal Distribution
799  @section The Lognormal Distribution  @section The Lognormal Distribution
# Line 787  using the formula given above. Line 829  using the formula given above.
829  \centerline{\input rand-lognormal.tex}  \centerline{\input rand-lognormal.tex}
830  @end tex  @end tex
831    
832    @deftypefun double gsl_cdf_lognormal_P (double @var{x}, double @var{zeta}, double @var{sigma})
833    @deftypefunx double gsl_cdf_lognormal_Q (double @var{x}, double @var{zeta}, double @var{sigma})
834    @deftypefunx double gsl_cdf_lognormal_Pinv (double @var{P}, double @var{zeta}, double @var{sigma})
835    @deftypefunx double gsl_cdf_lognormal_Qinv (double @var{Q}, double @var{zeta}, double @var{sigma})
836    These functions compute the cumulative distribution functions
837    @math{P(x)}, @math{Q(x)} and their inverses for the lognormal
838    distribution with parameters @var{zeta} and @var{sigma},
839    @end deftypefun
840    
841    
842  @page  @page
843  @node The Chi-squared Distribution  @node The Chi-squared Distribution
844  @section The Chi-squared Distribution  @section The Chi-squared Distribution
# Line 848  the formula given above. Line 900  the formula given above.
900  @deftypefunx double gsl_cdf_chisq_Pinv (double @var{P}, double @var{nu})  @deftypefunx double gsl_cdf_chisq_Pinv (double @var{P}, double @var{nu})
901  @deftypefunx double gsl_cdf_chisq_Qinv (double @var{Q}, double @var{nu})  @deftypefunx double gsl_cdf_chisq_Qinv (double @var{Q}, double @var{nu})
902  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
903  @math{P(x)} and @math{Q(x)}, and their inverses, for the chi-squared  @math{P(x)}, @math{Q(x)} and their inverses for the chi-squared
904  distribution with @var{nu} degrees of freedom.  distribution with @var{nu} degrees of freedom.
905  @end deftypefun  @end deftypefun
906    
# Line 986  given above. Line 1038  given above.
1038  @deftypefunx double gsl_cdf_tdist_Pinv (double @var{P}, double @var{nu})  @deftypefunx double gsl_cdf_tdist_Pinv (double @var{P}, double @var{nu})
1039  @deftypefunx double gsl_cdf_tdist_Qinv (double @var{Q}, double @var{nu})  @deftypefunx double gsl_cdf_tdist_Qinv (double @var{Q}, double @var{nu})
1040  These functions compute the cumulative distribution functions  These functions compute the cumulative distribution functions
1041  @math{P(x)} and @math{Q(x)}, and their inverses, for the t-distribution  @math{P(x)}, @math{Q(x)} and their inverses for the t-distribution
1042  with @var{nu} degrees of freedom.  with @var{nu} degrees of freedom.
1043  @end deftypefun  @end deftypefun
1044    
# Line 1069  formula given above. Line 1121  formula given above.
1121  \centerline{\input rand-logistic.tex}  \centerline{\input rand-logistic.tex}
1122  @end tex  @end tex
1123    
1124    @deftypefun double gsl_cdf_logistic_P (const double @var{x}, const double @var{a})
1125    @deftypefunx double gsl_cdf_logistic_Q (const double @var{x}, const double @var{a})
1126    @deftypefunx double gsl_cdf_logistic_Pinv (const double @var{P}, const double @var{a})
1127    @deftypefunx double gsl_cdf_logistic_Qinv (const double @var{Q}, const double @var{a})
1128    These functions compute the cumulative distribution functions
1129    @math{P(x)}, @math{Q(x)} and their inverses for the logistic
1130    distribution with scale parameter @var{a}.
1131    @end deftypefun
1132    
1133  @page  @page
1134  @node The Pareto Distribution  @node The Pareto Distribution
1135  @section The Pareto Distribution  @section The Pareto Distribution
# Line 1105  the formula given above. Line 1166  the formula given above.
1166  \centerline{\input rand-pareto.tex}  \centerline{\input rand-pareto.tex}
1167  @end tex  @end tex
1168    
1169    @deftypefun double gsl_cdf_pareto_P (double @var{x}, double @var{a}, double @var{b})
1170    @deftypefunx double gsl_cdf_pareto_Q (double @var{x}, double @var{a}, double @var{b})
1171    @deftypefunx double gsl_cdf_pareto_Pinv (double @var{P}, double @var{a}, double @var{b})
1172    @deftypefunx double gsl_cdf_pareto_Qinv (double @var{Q}, double @var{a}, double @var{b})
1173    These functions compute the cumulative distribution functions
1174    @math{P(x)}, @math{Q(x)} and their inverses for the Pareto
1175    distribution with exponent @var{a} and scale @var{b}.
1176    @end deftypefun
1177    
1178  @page  @page
1179  @node The Spherical Distribution (2D & 3D)  @node The Spherical Distribution (2D & 3D)
1180  @section The Spherical Distribution (2D & 3D)  @section The Spherical Distribution (2D & 3D)
# Line 1125  uniform random number between 0 and @mat Line 1195  uniform random number between 0 and @mat
1195  @var{y} be the sine and cosine respectively.  Two trig functions would  @var{y} be the sine and cosine respectively.  Two trig functions would
1196  have been expensive in the old days, but with modern hardware  have been expensive in the old days, but with modern hardware
1197  implementations, this is sometimes the fastest way to go.  This is the  implementations, this is sometimes the fastest way to go.  This is the
1198  case for my home Pentium (but not the case for my Sun Sparcstation 20 at  case for the Pentium (but not the case of a Sun Sparcstation 20).  
1199  work).  Once can avoid the trig evaluations by choosing @var{x} and  One can avoid the trigonometric evaluations by choosing @var{x} and
1200  @var{y} in the interior of a unit circle (choose them at random from the  @var{y} in the interior of a unit circle (choose them at random from the
1201  interior of the enclosing square, and then reject those that are outside  interior of the enclosing square, and then reject those that are outside
1202  the unit circle), and then dividing by @c{$\sqrt{x^2 + y^2}$}  the unit circle), and then dividing by @c{$\sqrt{x^2 + y^2}$}
# Line 1207  using the formula given above. Line 1277  using the formula given above.
1277  \centerline{\input rand-weibull.tex}  \centerline{\input rand-weibull.tex}
1278  @end tex  @end tex
1279    
1280    @deftypefun double gsl_cdf_weibull_P (double @var{x}, double @var{a}, double @var{b})
1281    @deftypefunx double gsl_cdf_weibull_Q (double @var{x}, double @var{a}, double @var{b})
1282    @deftypefunx double gsl_cdf_weibull_Pinv (double @var{P}, double @var{a}, double @var{b})
1283    @deftypefunx double gsl_cdf_weibull_Qinv (double @var{Q}, double @var{a}, double @var{b})
1284    These functions compute the cumulative distribution functions
1285    @math{P(x)}, @math{Q(x)} and their inverses for the Weibull
1286    distribution with scale @var{a} and exponent @var{b}.
1287    @end deftypefun
1288    
1289    
1290  @page  @page
1291  @node The Type-1 Gumbel Distribution  @node The Type-1 Gumbel Distribution
1292  @section  The Type-1 Gumbel Distribution  @section  The Type-1 Gumbel Distribution
# Line 1244  using the formula given above. Line 1324  using the formula given above.
1324  \centerline{\input rand-gumbel1.tex}  \centerline{\input rand-gumbel1.tex}
1325  @end tex  @end tex
1326    
1327    @deftypefun double gsl_cdf_gumbel1_P (double @var{x}, double @var{a}, double @var{b})
1328    @deftypefunx double gsl_cdf_gumbel1_Q (double @var{x}, double @var{a}, double @var{b})
1329    @deftypefunx double gsl_cdf_gumbel1_Pinv (double @var{P}, double @var{a}, double @var{b})
1330    @deftypefunx double gsl_cdf_gumbel1_Qinv (double @var{Q}, double @var{a}, double @var{b})
1331    These functions compute the cumulative distribution functions
1332    @math{P(x)}, @math{Q(x)} and their inverses for the Type-1 Gumbel
1333    distribution with parameters @var{a} and @var{b}.
1334    @end deftypefun
1335    
1336    
1337  @page  @page
1338  @node The Type-2 Gumbel Distribution  @node The Type-2 Gumbel Distribution
1339  @section  The Type-2 Gumbel Distribution  @section  The Type-2 Gumbel Distribution
# Line 1281  using the formula given above. Line 1371  using the formula given above.
1371  \centerline{\input rand-gumbel2.tex}  \centerline{\input rand-gumbel2.tex}
1372  @end tex  @end tex
1373    
1374    @deftypefun double gsl_cdf_gumbel2_P (double @var{x}, double @var{a}, double @var{b})
1375    @deftypefunx double gsl_cdf_gumbel2_Q (double @var{x}, double @var{a}, double @var{b})
1376    @deftypefunx double gsl_cdf_gumbel2_Pinv (double @var{P}, double @var{a}, double @var{b})
1377    @deftypefunx double gsl_cdf_gumbel2_Qinv (double @var{Q}, double @var{a}, double @var{b})
1378    These functions compute the cumulative distribution functions
1379    @math{P(x)}, @math{Q(x)} and their inverses for the Type-2 Gumbel
1380    distribution with parameters @var{a} and @var{b}.
1381    @end deftypefun
1382    
1383    
1384  @page  @page
1385  @node General Discrete Distributions  @node General Discrete Distributions
1386  @section General Discrete Distributions  @section General Discrete Distributions

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