Justify or prove all answers.

(20 points) Consider a bus stop where buses arrive according to a Poisson process with rate b per hour. Passengers arrive according to a Poisson process (independent of the arrival of the buses) at a rate of p per hour. Every bus arrives empty and has a capacity of 40 passengers. Passengers form a line to wait and board the buses as they arrive. (a) Suppose the bus stop is empty now. What is the probability that no passengers arrive in the next hour? (b) Suppose there is exactly one passenger waiting now. What is the probability that the passenger will depart within the next hour? (Buses depart immediately after arrival even if they are not full.) (0) Suppose there is exactly one passenger waiting now. What is the probability that the passenger waits alone for the next 30 minutes (so no buses or passengers arrive)? (d) Suppose a passenger arrives now and there are already 49 people in line. What is the expected time this passenger waits to board a bus? (e) Suppose the bus stop has 10 passengers now. What is the probability that the next bus to arrive will pick up a full load (40 passengers)