Let the random variables (discrete) X1, X2 have the joint pmf p(x1, x2) = 1/3, for (x1, 12) = (-1,0), (0, 1), (1, 0) . %3D (a) Compute Cov(X1, X2). (b) Are X1 and X2 independent? Why?

2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX,Y) = a Cov(X,Y). You may find it helpful to remember that if EX = µx, then EaX = aux. (b) Show that Cov(X + b, Y) = Cov(X,Y). Now let X, Y, Z be independent random variables with common variance o?. (c) Find the value of Corr(2X – 3Y + 4, 2Y – Z – 1). You may use any facts about covariance from - the notes, including those from parts (a) and (b) of this question, provided you state them clearly.

6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.

Let W = 2X + 5Y – 3Z, where the random variables X, Y, and Z satisfy E(X)= 2, V(X)= 5, E(Y)= -3, V(Y)=4, E(Z)=7, V(Z) = 6, Cov(X,Y) = -1, Cov(Y, Z) =3, Cov(Z, X)= 2. Calculate E(W), V (W) and Cov(W, Y).