I Let X, X2,. , Xn be a random sample from a population with probability mass func- tion given by f(x) = S0/2 1-0 0 if x = 0,1 if x = 2 otherwise 2 where (0, 1) is an unknown parameter. Define N(X) as the number of X's that result in the value 2. Also define the statistics T(X) (4 - 2X)/3 and T(X) 1 - N(X)/n. = = (i) Show that E(T;(X)) = 0, j = 1, 2, that is, T(X) and T(X) are unbiased estima- tors of 0. (ii) Show that N(X) is a minimal sufficient statistic for 0. (iii) Is N(X) a complete statistic? Explain your answer. (iv) Show that X is not sufficient for 0. (v) Compute and compare the variances Var(T (X)) and Var(T(X)).