Problem 1 (30 points). Prove the following claims and show your calculations. (a) Prove that EY]

Problem 1 (30 points). Prove the following claims and show your calculations. (a) Prove that E[Y] E[Y2] 1/2 for any real random variable Y. Moreover, show this implies E Var(X). (b) For n independent variables Xl, ... , n, prove that Var(X1 + X2 -4- -4- Xn) = Var(X1) -4- Var(X2) + + Var(Xn). (c) Consider d independent random coins Zl, ... , Zd G {1} where each Zi is 1 or 1 with probability 1/2 separately. We define n = 2d 1 random variables as follows: For each non-empty subset S C [n], we define Xs = Z For example when d = 3, there are 7 random variables Xl Zl,X2 = Z2,X3 = 4, X12 4 4, and 4. Z2.Z3. Calculate Var(Es X s) and compare this with part (b).