Let X1,..., Xn be i.i.d. Gamma(a, 1/a) random variables for some a > 0. We recall that the corresponding density is given by fa(x) = -x (here, I(s)-1 e* dx). = 1 T(a)aa -1 " -29-1-x/a\/x > 0 N (a) What is the limit, in probability, of the sample average X of the sample? (b) Find a function g such that g(Xn) is a consistent estimator of a. Denote by this estimator. (c) Prove that is asymptotically normal. What is its asymptotic variance ? (d) Using the previous question, find a confidence interval for a with asymptotic level 90%.