ABSTRACT
The probability density function of a noncentral chi-square random variable with the degrees of freedom n and the noncentrality parameter δ is given by
f(x|n, δ) = ∞∑ k=0
2 n+2k 2 Γ(n+2k2 )
, (17.1.1)
where x > 0, n > 0, and δ > 0. This random variable is usually denoted by χ2n(δ). It is clear from the density function (17.1.1) that conditionally given K, χ2n(δ) is distributed as χ
2 n+2K , where K is a Poisson random variable with mean
δ/2. Thus, the cumulative distribution of χ2n(δ) can be written as
P (χ2n(δ) ≤ x|n, δ) = ∞∑ k=0
P (χ2n+2k ≤ x). (17.1.2)
The plots of the noncentral chi-square pdfs in Figure 17.1 show that, for fixed n, χ2n(δ) is stochastically increasing with respect to δ, and for large values of n, the pdf is approximately symmetric about its mean n+ δ.
