For n 1, let X1, X2, Xn be a random sample (that is, X1, X2,..., X, are inde- pendent) from a geometric distribution with success probability p = 0.8. (a) Find the mgf My, (t) of Ys = X+X2+ X3+ X+Xs using the geometric mgf. Then name the distribution of Y, and give the value of its parameter(s). (b) Find the mgf My (t) of Y, X + X++ X, for any 21. Then name the distribution of Y, and give the value of its parameter(s). (c) Find the mgf Me (t) of the sample mean Y = Ya, " For the next two questions, Taylor series expansion of ea and the result lim [1+an+o(n1)]bm=eab may be useful. 8111 (d) Find the limit lim, My, (t) using the result of (c). What distribution does the limiting mgf correspond to? (e) Let Y Z = n =5nY - 5n. Find M, (t), the mgf of Z. Then use a theoretical argument to find the limiting, mgf lim, Mz(t). What is the limiting distribution of Z