BETNDF
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, and LAMBDA = 10.
 
USE UMACH_INT
USE BETNDF_INT
USE FNDF_INT
IMPLICIT NONE
INTEGER NOUT, I
REAL X, LAMBDA, SHAPE1, SHAPE2, &
BCDFV, FCDFV, F(8)
 
DATA F /0.0, 0.4, 0.8, 1.2, &
1.6, 2.0, 2.8, 4.0 /
 
CALL UMACH (2, NOUT)
SHAPE1 = 50.0
SHAPE2 = 5.0
LAMBDA = 10.0
 
WRITE (NOUT,'(/" SHAPE1: ", F4.0, &
& "; SHAPE2: ", F4.0, &
&"; LAMBDA: ", F4.0 // &
& 6x,"X",6x,"NCBETCDF(X)",3x,"NCBETCDF(X)"/ &
& 14x,"expected")') SHAPE1, SHAPE2, LAMBDA
 
DO I = 1, 8
X = (SHAPE1*F(I)) / (SHAPE1*F(I) + SHAPE2)
FCDFV = FNDF(F(I),2*SHAPE1,2*SHAPE2,LAMBDA)
BCDFV = BETNDF(X, SHAPE1, SHAPE2, LAMBDA)
WRITE (NOUT,'(2X, F8.6, 2(2X, E12.6))') &
X, FCDFV, BCDFV
END DO
END
Output
 
SHAPE1: 50.; SHAPE2: 5.; LAMBDA: 10.
 
X NCBETCDF(X) NCBETCDF(X)
expected
0.000000 0.000000E+00 0.000000E+00
0.800000 0.488790E-02 0.488790E-02
0.888889 0.202633E+00 0.202633E+00
0.923077 0.521143E+00 0.521143E+00
0.941176 0.733853E+00 0.733853E+00
0.952381 0.850413E+00 0.850413E+00
0.965517 0.947125E+00 0.947125E+00
0.975610 0.985358E+00 0.985358E+00