Question 2 A random variable X is said to be distributed according to a Gamma distri- bution if it has a pdf with the form -xx- -Bx e f(x)= = F(a) if x > 0 0 otherwise. with parameters a > 0, > 0. Remark: the Gamma distribution with a = n and = 1 is a special case. a) Derive its MGF. b) Use the MGF to derive the first 2 moments. c) Identify the distribution of cX by deriving its MGF. d) Identify the distribution of the sum of two independent Gamma random variables X1 2 Gamma(a1, B), X2 ~ Gamma (a2, 3) by deriving the MGF of X1 X2. e) With X and X2 as defined in part d, identify the distribution of X1+X2 in another way without using the MGF.