Definition
The probability density function is given by
where is distributed as chi-squared with degrees of freedom.
From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable J has a Poisson distribution with mean, and the conditional distribution of Z given is chi-squared with k+2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter .
Alternatively, the pdf can be written as
where is a modified Bessel function of the first kind given by
Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:
Siegel (1979) discusses the case k=0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.
Read more about this topic: Noncentral Chi-squared Distribution
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