Let X1, X2, ..., Xn be a collection of independent random variables with E( X;) = u and Var(X;) = o' for each i = 1,...,n. Let = W1,...,Wn be weights that add to one; that is >wi = 1. Consider the weighted sum Zn = El- wiXi. (a) What is E(Zn)? (b) What is Var(Zn)? (c) What do parts (a) and (b) tell us about the sample average Xn? (d) Does Zn converge in probability to E( Zn)? If yes, provide reasoning. If not, find a necessary condition that ensures the convergence in probability