Jill arrives at a restaurant at 6PM and waits for her three friends Moe, Agnes, and Dorothy. Each one of her friends arrives independently at a time uniformly distributed between 6PM and 7PM. (a) Find the expected value of the time at which Moe arrives. (b) Find the PDF of the time T at which ALL three friends have arrived. (Hint: find the CDF of T and use independence of the arrivals). (c) Find the probability that Jill has to wait more than half an hour by herself. (d) Find the PDF of T given that Moe arrives at 6:20 and Agnes arrives at 6:40.

Acaterer knows he will need 60, 50, 80, 40 and 50 dinner napkins on five successive evenings. he can purchase new napkins initially at 25 cents each, after which he can have dirty napkins laundered by a fast one-day laundry service (i.e., dirty napkins given at the end of the day will be ready for use the following day) at 15 cents each, or by a slow two-day service at 8 cents each or both. the caterer wants to know how many napkins he should purchase initially and how many dirty napkins should be laundered by fast and slow service on each of the days in order to minimize his total costs. formulate the caterer’s problem as a linear program as follows (you must state any assumptions you make): a. define all variables clearly. how many are there? b. write out the constraints that must be satisfied, briefly explaining each. (do not simplify.) write out the objective function to be minimized. (do not simplify.)