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

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 2.41 to show that the density function of n

i=1 Z2 i is

Transcribed Image Text:

e-x/2xn/2-1 f(x)= x>0 2n/2 (n/2)