1) Alice and Betty enter a beauty parlor simultaneously, Alice to get a manicure and Betty toget a haircut. Suppose the time for a manicure (haircut) is exponentially distributed withmean 20 (30) minutes.
(a) What is the probability Alice gets done first?.
(b) What is the expected amount of time until Alice and Betty are both done?
2) Customers arrive at a shipping office at times of a Poisson process with rate 3 per hour. Theoffice was supposed to open at 8AM but the clerk Oscar overslept and came in at 10AM.
(a) What is the probability that no customers came in the two-hour period?
(b) What is the distribution of the amount of time Oscar has to wait until his first customerarrives?
4) The number of hours between successive trains is T which is uniformly distributed between 1and 2. Passengers arrive at the station according to a Poisson process with rate 24 per hour.Let X denote the number of people who get on a train. Find
(a) E[X]
(b) Var(X)
5) Customers arrive at a sporting goods store at rate 10 per hour. 60% of the customers are menand 40% are women. Women spend an amount of time shopping that is uniformly distributedon [0, 30] minutes, while men spend an exponentially distributed amount of time with mean30 minutes. Let M and N be the number of men and women in the store. What is thedistribution of (M, N) in the long run?
6) Signals are transmitted according to a Poisson process with rate λ. Each signal is successfullytransmitted with probability p and lost with probability 1?p. The fates of different signalsare independent. For t ≥ 0 let N1(t) be the number of signals successfully transmitted andlet N2(t) be the number that are lost up to time t.
(a) Find the distribution of (N1(t), N2(t)).
(b) What is the distribution of L = the number of signals lost before the first one is successfullytransmitted?