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1. (12 marks) Let X and Y be two independent N(0, 1) random variables, and U = X +Y. (a) Find the conditional PDF

1. (12 marks) Let X and Y be two independent N(0, 1) random variables, and U = X +Y. (a) Find the conditional PDF of U given X = x, fu|X(u|x) . (b) Find the marginal PDF of U, fu(u). . (c) Find the conditional PDF of X given U = u, fx|U(x|u). (d) Find E[X|U = u] and Var(X|U = u).

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