Let X, X2,..., Xn (n> 1) be iid random variables having the Normal distribution with mean and variance o2. We define: S = X = X + X3 + X + X + n Answer the following questions: (a) Find the distribution of S, with parameters. We define the random variables X and Y as follows: Note that X = X and X = Y. X = X, Y = n Xn + Xn 1 X +=S. n [3 marks] (b) Find E(X), Var(X), E(Y), Var(Y), Cov(X, Y) and PX,Y. [6 marks] (c) Using the fact that the random variables X and S are independent, name the joint distribution of (X, Y). Do not use calculus. [1 mark] (d) Apply the results that you have obtained in parts (b) and (c) in order to find the conditional distribution of (X|X), with parameters. Do not use calculus. [3 marks] (e) Calculate E(X|X). Compare your result with E(X X). [4 marks] (f) In the case when n = 2, calculate Var(X|X). Compare your result with Var(X X). [5 marks]