ABSTRACT
The probability density function (pdf) 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
exp (− δ
) ( δ 2
)k k!
exp (−x
) x n+2k
2 n+2k
) , (18.1)
where x > 0, n > 0, and δ > 0. This random variable is usually denoted by χ2n(δ). It is clear from the density function (18.1) that conditionally given K, χ2n(δ) is distributed as χ2n+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
exp (− δ
) ( δ 2
)k k!
