1. [Random Vectors] Let X and Z are independent and identically distributed discrete random variables with support Sx = {−1,1} and probability mass function (PMF) is given by 0.5, x = 1 TX (x): = 0.5, x = -1 There given other random variable, Y, that is defined as Y = (a) Characterize support and PMF for Y. 1, if X = 1 Z, if X = −1 (b) Find variance-covariance matrix for random vector (X,Y). Are X and Y uncorrelated? (c) Find the optimal (in the mean squared error sense) predictor of Y given X, E(Y|X). (d) Find the optimal (in the mean squared error sense) linear predictor of Y given X. This implies finding the coefficients a and b from g(X) = a+b. X, such that E(Y - 9(X))² is minimized.

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