1. Let X = (X1, X2, . .., Xn) be a random sample of size n from the density f (I; 0) = 0(1 - 0)3, = {0, 1, 2, . . . }, Here 0 E (0, 1) is an unknown parameter. The mean and variance of X are given respectively by E(X) = 1 - 0 and Var(X) = 1- 0 02 a) Use any argument to prove that T = _? X, is a complete and minimal suffi- cient statistic for 0. b) Show and argue that the expected Fisher information about 6 contained in the statistic found in a) is n IT(0) = 02 ( 1 - 0 ) R c) Derive the UMVUE of h(0) = 0. Simply your UMVUE as much as possible. Hint: Use the interpretation that P(X1 = 0) = 0 and the fact that n T = X; ~ Negative Binomial(n, () i= with probability mass function P(T = t) = t (n + t - 1 ) on (1 - 0), t= 0, 1,2