This function evaluates the noncentral beta probability density function.
Function Return Value
BETNPR — Function value, the value of the probability density function. (Output)
Required Arguments
X — Argument for which the noncentral beta probability density 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: BETNPR (X, SHAPE1, SHAPE2, LAMBDA)
Specific: The specific interface names are S_BETNPR and D_BETNPR.
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 PDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F PDF:
where is a noncentral beta PDF with x = x, α1 = SHAPE1, α2 = SHAPE2, and noncentrality parameter λ = LAMBDA; is a noncentral F PDF with argument f , numerator and denominator degrees of freedom 2α1 and 2α2 respectively, and noncentrality parameter λ; and:
(See the documentation for function FNPR for a discussion of how the noncentral F PDF 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.
