. Let X and Y have the following joint distribution: X\\Y 0 1 0 0.40 0.10 0.10 0.10 2 0.10 0.20 (a) Find E[X|Y = 0]. (b) Find E[X|Y = 1] (c) What do your answers from (a) and (b) imply about the independence of X and Y? Explain your reason- ing. (d) Calculate the covariance between X and Y. (e) Find 17% and 0.21/- . Using the same joint distribution in problem 1; (a) Find Cor/(4 + 2X, 3 2Y). (b) Let Z = 3X 21' + 2. Find E[Z] and 523. (c) Calculate the correlation coefcient between X and Y. What does this suggest about the relationship between X and Y? Be as specic as you can. (d) Show that for two nonzero constants a. and b, Cor/(X + a, Y + b) = Cor/(X, Y). . Let X and Y be dependent random variables, where E[X] = 4,17X = 3,E[Y] = 2,ory = 6. Let p(X,Y) = 0.50. Find the expected value and the variance of the following linear combinations. (a) W1 = 12X+2Y+2 (b) W2 = (1/4)X 20Y + 100 (C) W3 = X Y ((1) W4 = X + Y . Let X and Y have the following joint distribution: X\\Y 0 1 2 0 5/50 8/50 1/50 2 10/50 1/50 5/50 4 10/50 10/50 0 /1664 /3111 Further, suppose ox = W103\" = m. (a) Find Cor/(X, Y). (b) Find p(X, Y). (c) Find Cor/(1 X, 10 + Y) (A) Find n{1_ Y 104V). Hie-1+ Mn\" chaos/Tn? amp {pl and