gamma_cdf

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.