chapter 18

8 Pages

## - Noncentral Chi-square Distribution

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!