42. If Xi , i = 1, . . . , n are independent normal random variables, with Xi having mean

μi and variance 1, then the random variable

ni

=1 X2 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|

(b) Derive the moment generating function of the noncentral chi-squared random variable ni =1 X2 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 ni =1 X2 i is a noncentral chi-squared random variable with parameters n and θ = ni =1 μ2i .

(c) If all μi = 0, then ni =1 X2 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.

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