GAMDF
This function evaluates the gamma cumulative distribution function.
Function Return Value
GAMDF — Function value, the probability that a gamma random variable takes a value less than or equal to X. (Output)
Required Arguments
X — Argument for which the gamma distribution function is to be evaluated. (Input)
A — The shape parameter of the gamma distribution. (Input)
This parameter must be positive.
This parameter must be positive.
FORTRAN 90 Interface
Generic: GAMDF (X, A)
Specific: The specific interface names are S_GAMDF and D_GAMDF.
FORTRAN 77 Interface
Single: GAMDF (X, A)
Double: The double precision name is DGAMDF.
Description
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.)
Figure 28, Gamma Distribution Function
Comments
Informational error
Type | Code | Description |
|---|---|---|
1 | 2 | Since the input argument X is less than zero, the distribution function is set to zero. |
Example
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.
USE UMACH_INT
USE GAMDF_INT
IMPLICIT NONE
INTEGER NOUT
REAL A, P, X
!
CALL UMACH (2, NOUT)
A = 4.0
X = 0.5
P = GAMDF(X,A)
WRITE (NOUT,99998) P
99998 FORMAT (' The probability that X is less than 0.5 is ', F6.4)
X = 1.0
P = GAMDF(X,A) - P
WRITE (NOUT,99999) P
99999 FORMAT (' The probability that X is between 0.5 and 1.0 is ', &
F6.4)
END
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
