Problem 9.3 (Video 7.1, 7.2, 8.1, 8.2, Quick Calculations) For each of the scenarios below, determine the requested quantities. (You should be able to do this without any long calculations or integration.) (a) Assume that X is Uniform(1, 3), V is Gaussian(1, 1), X and V are independent, and Y = X + V.

Determine the LLSE estimator of X given Y = y and the corresponding mean squared error. 1 (b) Let X1, X2, . . . , X400 be a collection of continuous, independent, identically distributed continuous random variables, each of which is uniformly distributed over [5, 5]. Let Y = 1 400 P400 i=1(Xi) 3 . Compute E[Y ] and Var[Y ]. (c) Let X1, X2, . . . , X100 be a collection of continuous, independent, Exponential( 1 2 random variables. Let Y = 1 100 P100 i=1 Xi . Use the Central Limit Theorem to estimate P[Y 1.9].