This function evaluates the noncentral chi‑squared cumulative distribution function.

Function CSNDF evaluates the cumulative distribution function of a noncentral chi‑squared random variable with DF degrees of freedom and noncentrality parameter ALAM, that is, with ν = DF, λ = ALAM, and x = CHSQ,

where Γ(⋅) is the gamma function. This is a series of central chi‑squared distribution functions with Poisson weights. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

The noncentral chi‑squared random variable can be defined by the distribution function above, or alternatively and equivalently, as the sum of squares of independent normal random variables. If Yi have independent normal distributions with means μi and variances equal to one and

then X has a noncentral chi‑squared distribution with n degrees of freedom and noncentrality parameter equal to

With a noncentrality parameter of zero, the noncentral chi‑squared distribution is the same as the chi‑squared distribution.

Function CSNDF determines the point at which the Poisson weight is greatest, and then sums forward and backward from that point, terminating when the additional terms are sufficiently small or when a maximum of 1000 terms have been accumulated. The recurrence relation 26.4.8 of Abramowitz and Stegun (1964) is used to speed the evaluation of the central chi‑squared distribution functions.

In this example, CSNDF is used to compute the probability that a random variable that follows the noncentral chi‑squared distribution with noncentrality parameter of 1 and with 2 degrees of freedom is less than or equal to 8.642.