16. I throw two dice and record the scores S1 and S2. Let X be the sum S1+S2 and Y the difference S1 − S2.

(a) Suppose the dice are fair, so that the values 1, 2, . . . , 6 are equally likely. Calculate the mean and variance of both X and Y . Find all the values of x and y at which the probabilities P(X = x), P(Y = y) are each either greatest or least. Determine whether the random variables X and Y are independent.

(b) Now suppose the dice give the values 1, 2, . . . , 6 with probabilities p1, p2, . . . , p6 and q1, q2, . . . , q6, respectively. Write down the values of P(X = 2), P(X = 7), and P(X = 12). By comparing P(X = 7) with √P(X = 2)P(X = 12) and applying the arithmetic/geometric mean inequality,4 or otherwise, show that X cannot be uniformly distributed on the set {2, 3, . . . , 12}.