Evaluates the gamma distribution function.
Synopsis
#include <imsl.h>
float imsl_f_gamma_cdf (float x, float a)
The type double procedure is imsl_d_gamma_cdf.
Required Arguments
float x
(Input)
Argument for which the gamma distribution function is to be
evaluated.
float a
(Input)
The shape parameter of the gamma distribution. This parameter must be
positive.
Return Value
The probability that a gamma random variable takes a value less than or equal to x.
Description
The function imsl_f_gamma_cdf evaluates the distribution function, F, of a gamma random variable with shape parameter a, that is,
where Γ(×) is the gamma function. (The gamma function is the integral from zero to infinity of the same integrand as above). The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
The gamma distribution is often defined as a two-parameter distribution with a scale parameter b (which must be positive) or even as a three-parameter distribution in which the third parameter c is a location parameter.
In the most general case, the probability density function
over
(c, ¥) is
If T is such a random variable with parameters
a, b, and c, the probability that
T ≤ t0 can be obtained from
imsl_f_gamma_cdf
by setting x = (t0 − c)/b.
If x is less than a or if x is less than or equal to 1.0, imsl_f_gamma_cdf uses a series expansion. Otherwise, a continued fraction expansion is used. (See Abramowitz and Stegun 1964.)
Example
Let X be a gamma random variable with a shape parameter of four. (In this case, it has an Erlang distribution since the shape parameter is an integer.) This example finds the probability that X is less than 0.5 and the probability that X is between 0.5 and 1.0.
#include
<imsl.h>
main()
{
float p, x;
float a = 4.0;
x =
0.5;
p = imsl_f_gamma_cdf(x,a);
printf("The probability that X is less than 0.5 is %6.4f\n",
p);
x = 1.0;
p =
imsl_f_gamma_cdf(x,a) - p;
printf("The probability that X
is between 0.5 and 1.0 is %6.4f\n", p);
}
Output
The probability that X is less than 0.5 is 0.0018
The
probability that X is between 0.5 and 1.0 is 0.0172
Informational Errors
IMSL_LESS_THAN_ZERO The input argument, x, is less than zero.
Fatal Errors
IMSL_X_AND_A_TOO_LARGE The function overflows because x and a are too large.
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