87. Recall that X is said to be a gamma random variable with parameters (, ) if

87. Recall that X is said to be a gamma random variable with parameters (α, λ) if its density is f (x) = λe−λx (λx)

α−1/ (α), x > 0

(a) If Z is a standard normal random variable, show that Z2 is a gamma random variable with parameters (1/2, 1/2).

(b) If Z1,..., Zn are independent standard normal random variables, then n

i=1 Z2 i is said to be a chi-squared random variable with n degrees of freedom. Explain how you can use results from Example 2.39 to show that the density function of n i=1 Z2 i is f (x) = e−x/2xn/2−1 2n/2 (n/2)

, x > 0