2.20 Consider three random variables X, Y, and Z. Suppose that Y takes on k values y1,

c, yk, that X takes on l values x1,

c, xl, and that Z takes on m values z1,

c, zm. The joint probability distribution of X, Y, Z is Pr(X = x, Y = y, Z = z), and the conditional probability distribution of Y given X and Z is Pr(Y = y  X = x, Z = z) = Pr(Y = y, X = x, Z = z)

Pr(X = x, Z = z) .

a. Explain how the marginal probability that Y = y can be calculated from the joint probability distribution. [Hint: This is a generalization of Equation (2.16).]

b. Show that E(Y) = E[E(Y 0 X, Z)]. [Hint: This is a generalization of Equations (2.19) and (2.20).]