Evaluates the noncentral chi-squared probability density function.

The type double function is imsls_d_non_central_chi_sq_pdf.

The probability density associated with a noncentral chi-squared random variable with value x.

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), 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 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 imsls_f_non_central_chi_sq_pdf 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 imsls_f_non_central_chi_sq 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 λ = 40.