If Xi, i = 1, . . . , n are independent normal random variables, with Xi having mean i and variance 1, then the

If Xi, i = 1, . . . , n are independent normal random variables, with Xi having mean μi and variance 1, then the random variable

n i=1X2 i is said to be a noncentral chi-squared random variable.

(a) if X is a normal random variable having mean μ and variance 1 show, for

|t | < 1/2, that the moment generating function of X2 is

(b) Derive the moment generating function of the noncentral chi-squared random variable n i=1X2 i , and show that its distribution depends on the sequence of means μ1, . . . , μn only through the sum of their squares.

As a result, we say that n i=1X2 i is a noncentral chi-squared random variable with parameters n and θ =n i=1 μ2 i .

(c) If all μi = 0, then n i=1X2 i is called a chi-squared random variable with n degrees of freedom. Determine, by differentiating its moment generating function, its expected value and variance.

(d) Let K be a Poisson random variable with mean θ/2, and suppose that conditional on K = k, the random variable W has a chi-squared distribution with n + 2k degrees of freedom. Show, by computing its moment generating function, that W is a noncentral chi-squared random variable with parameters n and θ.

(e) Find the expected value and variance of a noncentral chi-squared random variable with parameters n and θ.