Evaluates the gamma distribution function.
Synopsis
#include <imsls.h>
float imsls_f_gamma_cdf (float x, float a)
The type double function is imsls_d_gamma_cdf.
Required Arguments
float x
(Input)
Argument for which the gamma distribution function is to be
evaluated.
float a
(Input)
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
Function imsls_f_gamma_cdf evaluates the distribution function, F, of a gamma random variable with shape parameter a,
where Γ(⋅) is the gamma function. (The gamma function is the integral from 0 to ∞ 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 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 as follows:
If T is a random variable with parameters a, b, and c, the probability that T ≤ t0 can be obtained from imsls_f_gamma_cdf by setting x = (t0 − c)/b.
If x is less than a or less than or equal to 1.0, imsls_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 <imsls.h>
#include <stdio.h>
int main()
{
float x = 0.5, a = 4.0, p;
p = imsls_f_gamma_cdf(x,a);
printf("The probability that X is less than "
"%3.1f is %6.4f\n", x, p);
x = 1.0;
p = imsls_f_gamma_cdf(x,a) - p;
printf("The probability that X is between 0.5 and "
"%3.1f is %6.4f\n", x, 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
IMSLS_ARG_LESS_THAN_ZERO Since “x” = # is less than zero, the distribution function is zero at “x.”
Fatal Errors
IMSLS_X_AND_A_TOO_LARGE Since “x” = # and “a” = # are so large, the algorithm would overflow.
