40. A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. A committee of two is selected at random to work on a problem. Let X be the number of mathematics majors and let Y be the number of computer science majors chosen.

a. Find the joint probability mass function p(x, y).

This generalizes the hypergeometric distribution studied in Section 3.6. Give the joint probability table showing all nine values, of which three should be 0.

b. Find the marginal probability mass functions by summing numerically. How could these be obtained directly? Hint: What are the univariate distributions of X and Y?

c. Find the conditional probability mass function of Y given X  x for x  0, 1, 2. Compare with the h(y; 2  x, 6, 10) distribution. Intuitively, why should this work?

d. Are X and Y independent? Explain.

e. Find , x  0, 1, 2. Do this numerically and then compare with the use of the formula for the hypergeometric mean, using the hypergeometric distribution given in part (c). Is E(YX  x) a linear function of x?

f. Find , x  0, 1, 2. Do this numerically and then compare with the use of the formula for the hypergeometric variance, using the hypergeometric distribution given in part (c).