This function evaluates the noncentral chi‑squared probability density function.

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 σ2i, 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), andλ, 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 k + 2i 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.

This example calculates the noncentral chi‑squared distribution for a distribution with 100 degrees of freedom and noncentrality parameter  = 40.