IDL Analyst Reference Guide: Probability Distribution Functions and Inverses

IMSL_GAMMACDF

The IMSL_GAMMACDF function evaluates the gamma distribution function.

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

If present and nonzero, double precision is used.

Discussion

The IMSL_GAMMACDF function evaluates the distribution function, F, of a gamma random variable with shape parameter a; that is:

where G(¥) 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 T £ t₀ 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