Welcome to the L3 Harris Geospatial documentation center. Here you will find reference guides and help documents.
﻿
>  Docs Center  >  IDL Reference  >  Advanced Math and Stats  >  IMSL_GAMMACDF

### IMSL_GAMMACDF

IMSL_GAMMACDF

The IMSL_GAMMACDF function evaluates the gamma distribution function.

The IMSL_GAMMACDF function 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 0 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, infinity) is as follows:

If T is such a random variable with parameters a, b, and c, the probability that Tt0 can be obtained from IMSL_GAMMACDF by setting x = (t0 – c ) / b.

If x is less than a or if x is less than or equal to 1.0, IMSL_GAMMACDF 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 4. (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.

`a = 4`
`x = .5`
`p = IMSL_GAMMACDF(x, a)`
`PM, p, Title = 'The probability that X is less ' + \$`
`  'than .5 is:'`
` `
`The probability that X is less than .5 is: 0.00175162`
` `
`x = 1`
`p = IMSL_GAMMACDF(x, a) - p`
`PM, p, Title = 'The probability that X is between .5 and 1 is:'`
` `
`The probability that X is between .5 and 1 is: 0.0172365`

## Syntax

Result = IMSL_GAMMACDF(X, A [, /DOUBLE])

## Return Value

The probability that a gamma random variable takes a value less than or equal to x.

## Arguments

### A

Shape parameter of the gamma distribution. This parameter must be positive.

### X

Argument for which the gamma distribution function is to be evaluated.

## Keywords

### DOUBLE (optional)

If present and nonzero, double precision is used.

## Errors

### Informational Errors

STAT_LESS_THAN_ZERO: Input argument, x, is less than zero.

### Fatal Errors

STAT_X_AND_A_TOO_LARGE: Function overflows because x and a are too large.

## Version History

 6.4 Introduced