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 1X = x, Y = y, Z = z2, and the conditional probability distribution of Y given X and Z is Pr1Y = y X = x, Z = z2 = Pr1Y = y, X = x, Z = z2 Pr1X = x, Z = z2 .

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 E1Y2 = E3E1YX, Z24. [Hint: This is a generalization of Equations (2.19) and (2.20).]