36. According to an article in the August 30, 2002, issue of the Chron. Higher Ed., 30% of rst-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road.

Choose two students at random, let X be the number of liberals, and let Y be the number of conservatives.

a. Using the multinomial distribution from Section 5.1, give the joint probability mass function p(x, y) of X and Y. Give the joint probability table showing all nine values, of which three should be 0.

b. Find the marginal probability mass functions by summing p(x, y) 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 Bin[2  x, .2/(.2  .5)] distribution. Why should this work?

d. Are X and Y independent? Explain.

e. Find for x  0, 1, 2. Do this numerically and then compare with the use of the formula for the binomial mean, using the binomial E1Y 0 X  x2 distribution given in part (c). Is a linear function of x?

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