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revision 1.1.2.2 by jstover, Thu Jan 9 15:47:34 2003 UTC revision 1.1.2.3 by jstover, Wed Jan 15 01:10:48 2003 UTC
# Line 10  The functions described in this section Line 10  The functions described in this section
10  * Cumulative Distribution Function Usage::  * Cumulative Distribution Function Usage::
11  * The gsl_cdf_result struct::  * The gsl_cdf_result struct::
12  * The Gaussian CDF::    * The Gaussian CDF::  
13    * The Gamma CDF::
14  * CDF Examples::  * CDF Examples::
15  * CDF References and Further Reading::  * CDF References and Further Reading::
16  @end menu  @end menu
# Line 115  Pr(Z<x) = @{1 \over \sqrt@{2 \pi \sigma^ Line 116  Pr(Z<x) = @{1 \over \sqrt@{2 \pi \sigma^
116  @end example  @end example
117  @end ifinfo  @end ifinfo
118    
119    @page
120    @node The Gamma CDF
121    @section The Gamma Cumulative Distribution Function
122    
123    @deftypefun int gsl_cdf_gamma_e ( double @var{x}, double @var{scale}, double @var{shape}, gsl_cdf_tail_t @var{tail}, gsl_cdf_result @var{*result})
124    @deftypefun double gsl_cdf_gamma ( double @var{x}, double @var{scale}, double @var{shape}, gsl_cdf_tail_t @var{tail})
125    @end deftypefun
126    
127    These functions compute the probability @math{Pr(Z<x)} for a gamma
128    random variable with scale parameter @var{scale} and shape parameter
129    @var{shape} if @math{tail=GSL_CDF_LOWER}, or @math{Pr(Z>x)} if
130    @math{tail=GSL_CDF_UPPER}. A random variable is said to follow the
131    Gamma distribution with scale parameter @math{alpha} and shape
132    parameter @math{beta} if
133    @end deftypefun
134    
135    @tex
136    \beforedisplay
137    $$
138    Pr(Z<x) = { 1 \over alpha^{beta} \gamma{beta}} int_0^x t^{beta-1} \exp
139    (t/alpha) dt
140    $$
141    @end tex
142    @ifinfo
143    @example
144    Pr(Z<x) = @{ 1 \over alpha^@{beta@} \gamma@{beta@}@} int_0^x t^@{beta-1@} \exp
145    (t/alpha) dt
146    @end example
147    @end ifinfo
148    
149    The Chi-square distribution with @math{n} degrees of freedom
150    is a special case of the Gamma distribution with shape parameter
151    @math{n/2} and scale parameter @math{2}.
152    
153    The Gamma distribution is sometimes called the Erlang distribution in
154    queueing theory.
155    
156  @node CDF Examples  @node CDF Examples
157  @section Examples  @section Examples
158    
# Line 180  Chance that Z is greater than 0.320000 i Line 218  Chance that Z is greater than 0.320000 i
218  Chance that Z is greater than 1.500000 is 0.066807.  Chance that Z is greater than 1.500000 is 0.066807.
219  @end example  @end example
220    
221    
222    @noindent
223    The following program computes different values from the
224    Gamma distribution with shape parameter 2.3 and scale parameter 1.2.
225    
226    @example
227    #include <math.h>
228    #include <gsl/gsl_cdf.h>
229    #include <gsl/gsl_cdf_result.h>
230    #include <gsl/gsl_errno.h>
231    int main()
232    @{
233      gsl_cdf_result result;
234      double y;
235      double shape = 2.3;
236      double scale = 1.2;
237      int rc = 0;
238    
239      for ( y = -.5; y < 5.3; y += .02 )
240        @{
241          rc = gsl_cdf_gamma_e ( y, scale, shape, GSL_CDF_UPPER, &result
242          );
243          if ( rc == GSL_SUCCESS )
244            @{
245              printf ( "%f\n",result.val);
246            @}
247        @}
248      return rc;
249    @}
250    
251    This program can be compiled and run in the same way as that of the
252    previous example.
253    @end example
254    
255  @node CDF References and Further Reading  @node CDF References and Further Reading
256  @section References and Further Reading  @section References and Further Reading
257  @noindent  @noindent

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