question as below

5. [12 marks] Consider n independent identically distribution (iid) observations from Gamma(a, #) distribution. a) Determine the cumulant generating function for a single observation from a Gamma(a, B). b) Compute the first and second derivative of the cumulant generating function. c) Find the saddlepoint, t. d) Evaluate Kx(t) and K*() and simplify as much as possible. e) Show that the first order saddlepoint approximation for the density of X is (noB) me(no)no-le in. Hint: The first order saddlepoint approximation is 1( I ) - 1 efnkx(1)-niz) f) Hence, determine the first order saddlepoint approximation for the density of the sum of n iid observations Y , Et, Xi from this distribution. Hint: Consider using the density transformation formula fr(y) = fx(=(#)) dy g) What is the approximate distribution of this saddlepoint approximation? Keep in mind the famous Stirling approximation of the Gamma function that states no (no) noe no ~ I'(na)