This function evaluates the noncentral chi‑squared probability density function.
Function Return Value
CSNPR — Function value, the value of the probability density function. (Output)
Required Arguments
X — Argument for which the noncentral chi‑squared probability density function is to be evaluated. (Input) X must be non‑negative.
DF — Number of degrees of freedom of the noncentral chi‑squared distribution. (Input) DF must be positive.
LAMBDA — Noncentrality parameter. (Input) LAMBDA must be non‑negative.
FORTRAN 90 Interface
Generic: CSNPR (X, DF, LAMBDA)
Specific: The specific interface names are S_CSNPR and D_CSNPR.
Description
The noncentral chi‑squared distribution is a generalization of the chi‑squared distribution. If {Xi} are k independent, normally distributed random variables with means μi and variances σi2, then the random variable:
is distributed according to the noncentral chi‑squared distribution. The noncentral chi‑squared distribution has two parameters: k which specifies the number of degrees of freedom (i.e. the number of Xi), and λ which is related to the mean of the random variables Xi by:
The noncentral chi-squared distribution is equivalent to a (central) chi-squared distribution with degrees of freedom, where i is the value of a Poisson distributed random variable with parameter λ/2. Thus, the probability density function is given by:
where the (central) chi‑squared PDF f(x, k) is given by:
where Γ(⋅) is the gamma function. The above representation of F(x, k, λ) can be shown to be equivalent to the representation:
Function CSNPR(X, DF, LAMBDA) evaluates the probability density function of a noncentral chi‑squared random variable with DF degrees of freedom and noncentrality parameter LAMBDA, corresponding to k = DF, λ=LAMBDA, and x=X.
Function CSNDF(X, DF, LAMBDA) evaluates the cumulative distribution function incorporating the above probability density function.
With a noncentrality parameter of zero, the noncentral chi‑squared distribution is the same as the central chi‑squared distribution.
Example
This example calculates the noncentral chi‑squared distribution for a distribution with 100 degrees of freedom and noncentrality parameter λ = 40.
USE UMACH_INT
USE CSNPR_INT
IMPLICIT NONE
INTEGER :: NOUT, I
REAL :: X(6)=(/ 0.0, 8.0, 40.0, 136.0, 280.0, 400.0 /)
