This function evaluates the gamma cumulative distribution function.

Function GAMDF 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 ∞ 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, ∞) is

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

If X is less than a or if X is less than or equal to 1.0, GAMDF uses a series expansion. Otherwise, a continued fraction expansion is used. (See Abramowitz and Stegun, 1964.)

Suppose X is 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.) In this example, we find the probability that X is less than 0.5 and the probability that X is between 0.5 and 1.0.