77. Let Y1 and Yn be the smallest and largest order statistics, respectively, from a random sample of size n, and let W2  Yn  Y1 (this is the sample range).

a. Let W1  Y1, obtain the joint pdf of the Wi s (use the method of Section 5.4), and then derive an expression involving an integral for the pdf of the sample range.

b. For the case in which the random sample is from a uniform (0, 1) distribution, carry out the integration of

(a) to obtain an explicit formula for the pdf of the sample range.

a. Compute the marginal pmf s of X and Y.

b. What is the probability that the man s and the woman s dinner cost at most $15 each?

c. Are X and Y independent? Justify your answer.

d. What is the expected total cost of the dinner for the two people?

e. Suppose that when a couple opens fortune cookies at the conclusion of the meal, they nd the message You will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen.
How much does the restaurant expect to refund?