Question 5 [15 marks] a. Let U and V be independent continuous random variables. Show that the moment generating function (mgf) of W. W = U -V, can be obtained from those of U and V by Mw(t) = Mu(t)Mv(-t). b. If X and Y are independent Uniform (0, 1) random variables, show that the mef of Z, Z = X- Y, is Mz(t) = =( ettert- 2), 170. 1 , 1=0. c. By verifying that lim Mz(t) = Mz(0) = 1, 1-+0 show that the mgf in part (b) is continuous at * = 0. d. Find the kth raw moment , of Z by series expansion of Mz(t). e. Let Z = 1EL, Zi, where Z1,..., Zn are independent and identically distributed with common mgf My(t) given in part (b). What are the cumulants of the distri- bution of Z in terms of the cumulants &; of the distribution of Z? Note: it is NOT necessary to evaluate the cumulants of the distribution of Z