Let X = (X1, X2)', and X1 and X2 are iid random variables with common pmf f (x\0) xlog(1 – 0)' where x = 1,2,.., and 0 € (0, 1). a) Find the mean and variance of X1. b) Show that U(X) = -I(X1 = 1) is an unbiased estimator of 1(0) = 01 log(1– 0). Here, I(-) is an indicator function. c) Show that T(X) = X1 + X2 is a complete sufficient statistic for 0. d) Find the UMVUE of 1 (0).

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