(a) Consider a sequence of random variables (Y : i E N) taking values in (0, 1), and consider also a sequence of biased coins, where the ith coin takes the number that Y, generates as the probability of a head. Let S, be the number of heads after the flips of the first n coins. Are the assumptions of the version of the central limit theorem (CLT) provided in the lecture satisfied in the following scenarios? If so, provide the statement of the CLT with notations clearly defined. (i) Yi are independent and identically distributed; (ii) Y; = Y for all i, where Y is a random variable taking values in (0, 1). (b) Let (X1, . .., Xn ) be a random iid sample where X1 is uniformly distributed in {1, 2, . .., 0}, where 0 E Z+ is unknown. Let T - max{ X1, . . ., Xn}. Show that Tntl - (T - 1)n+1 W = TR - (T - 1)n is an unbiased and consistent estimator for 0. Hint: It might be helpful to use Th - (T - 1)" = [1 - (1,1) " IT"