The following table gives pxynbspx y the joint probability

Question

The following table gives P_X,Y (x, y), the joint probability mass function of random variables X and Y.

(a) Find the marginal probability mass functions P_X (x) and P_Y (y).

(b) Are X and Y independent?

(c) Find E[X], Var[X], E[Y], Var[Y], and Cov[X, Y].

(d) Let (Y) = aY + b be a linear estimator of X. Find a* and b*, the values of a and b that minimize the mean square error e_L.

(e) What is e∗_L, the minimum mean square error of the optimum linear estimate?

(f) Find P_X|Y (x| - 3), the conditional PMF of X given Y = -3.

(g) Find M (-3), the optimum (nonlinear) mean square estimator of X given Y = -3.

(h) What is

The mean square error of this estimate?