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.

Suppose that X₁ = 1; X₂ = 1; X3 = 0; X₁ = 1; X5 = 1, X6 = 1; X₂ = 0; X = 1; X9 = 0, X10 = 0, represents a random sample. Each of these X's comes from the same population and has a density of fx₁ = 0¹-* (1 - 0)*; x = 0,1 First determine the form of the maximum likelihood estimator for 0, then use the MLE formula and the data provided to find an estimate for 0. Round your answer to 3 decimal places. (Answer is 0.4)

Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)