4. (25 marks) Let X1, X2, ..., X, be a random sample from Unif[e - }, 0 + ;]. (a) (5 marks) Let X = EN-, X;. Show that X is unbiased and consistent. (b) (5 marks) Let Y = (X() +X(m))/2, where X() = min(X1, ..., X,,) and X(m) = max(X1, ..., X,.). Is Y an unbiased estimator for 0? Explain in details. (c) (5 marks) Find the variance of Y. (d) (5 marks) Consider the two estimators X and Y, which is more efficient? (e) (5 marks) Find a sufficient statistic for 0