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# Question 4 Suppose that a fisherman catches fish according to a Poisson process with rate 2 per hour. We know that

yesterday he began fishing at 9:00 pm. (a) What is the probability that he caught exactly two fish by 10:00 am yesterday? (b) What is the probability that he caught exactly two fish by 10:00 am yesterday and five fish by 11:00 am yesterday? (c) What is the expected time that he caught his fifth fish? (d) Suppose that we know that by 1:00 pm yesterday he caught exactly three fish. What is the probability that by 2:00 pm yesterday he caught a total of exactly ten fish? And what is the probability that he caught his first fish after 10:00 am yesterday? (e) Suppose that we know that he didn't catch any fish until after 10:00 am yesterday. Given that information, what is the expected time that he caught his first fish?

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