At a subway station, eastbound trains and northbound trains arrive independently, both according to a Poisson process. On average, there is one eastbound train every 12 minutes and one northbound train every 8 minutes. Suppose you arrive at the subway station at a certain point of time and start observing trains.
(a) What is the probability that exactly two eastbound trains will arrive in the first 24 minutes?
(b) What is the probability that exactly two eastbound trains will arrive in the first 24 minutes and exactly three eastbound trains will arrive in the first 36 minutes?
(c) What is the expected waiting time, in minutes, until the first train (of either type) arrives?
(d) What is the probability that the first train to arrive is eastbound, and the next two are northbound?
(e) What is the probability that it will take at least 20 minutes for two northbound trains to arrive?