Poisson process question attached below - please answer all parts

PROBLEM (a) (b) (C) (d) (e) (0 Suppose that customers arrive at a bank for one of two purposes: Either to deposit money or to withdraw money. Further, suppose that I __BANK_ i, depositing customers arrive at random times according to a Poisson i. I process with rate 7L, while withdrawing customers arrive independently |M a I according to a Poisson process with rate 1.1. 111 .! l \\ Find the probability that the next arrival is withdrawing. Find the probability that the second arrival is depositing. {The first could be anything.) Find the probability that the next three arrivals are all depositing. Given that three customers arrived in a given 90-second time window, what is the probability that two were depositing and one was withdrawing? The bank opens at 12 noon. Given that ve customers are known to have arrived in the 60-minute window between 12 noon and 1pm, what is the expected arrival time of the rst customer? Given that ve customers are known to have arrived in the 60-minute window between 12 noon and 1pm, what is the probability that all ve of them arrived in the 15-minute window from 12:45pm to 1pm