Answer.

(i) 2 points Assume X, X1, X2, ... is a sequence of real-valued random variables, defined on a common probability space (0, F, P), where, for n E N, Xn := 1ER] Xx is the associated sample mean. A. if Xn - X in probability as n -> co, then Xn - X almost surely as n - co; B. if E[Xn] = 2 and Var(Xn) = 3 for all n E N, Xn -+ 2 as n - 0%; C. if E[X?] + Ex, E[X2]