This function evaluates the noncentral beta cumulative distribution function (CDF).
Function Return Value
BETNDF — Probability that a random variable from a beta distribution having shape parameters SHAPE1 and SHAPE2 and noncentrality parameter LAMBDA will be less than or equal to X. (Output)
Required Arguments
X — Argument for which the noncentral beta cumulative distribution function is to be evaluated. (Input) X must be non‑negative and less than or equal to 1.
SHAPE1 — First shape parameter of the noncentral beta distribution. (Input) SHAPE1 must be positive.
SHAPE2 — Second shape parameter of the noncentral beta distribution. (Input) SHAPE2 must be positive.
LAMBDA — Noncentrality parameter. (Input) LAMBDA must be non‑negative.
FORTRAN 90 Interface
Generic: BETNDF (X, SHAPE1, SHAPE2, LAMBDA)
Specific: The specific interface names are S_BETNDF and D_BETNDF.
Description
The noncentral beta distribution is a generalization of the beta distribution. If Z is a noncentral chi‑square random variable with noncentrality parameter λ and 2α1 degrees of freedom, and Y is a chi‑square random variable with 2α2 degrees of freedom which is statistically independent of Z, then
is a noncentral beta‑distributed random variable and
is a noncentral F‑distributed random variable. The CDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F CDF:
where is a noncentral beta CDF with x = x, α1 = SHAPE1, α2 = SHAPE2, and noncentrality parameter λ = LAMBDA; is a noncentral F CDF with argument f, numerator and denominator degrees of freedom 2α1 and 2α2 respectively, and noncentrality parameter λ and:
(See documentation for function FNDF for a discussion of how the noncentral F CDF is defined and calculated.)
With a noncentrality parameter of zero, the noncentral beta distribution is the same as the beta distribution.
Example
This example traces out a portion of a noncentral beta distribution with parameters SHAPE1 = 50, SHAPE2 = 5, andLAMBDA = 10.
