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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 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.)

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