# rphw5solf - Stochastic Processes nctuee07f Homework 5...

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Stochastic Processes nctuee07f Homework 5 Solutions 1. Suppose the number of fish that a fisherman catches follows a Poisson process with rate λ = 0 . 5 per hour. The fisherman will keep fishing for two hours. If he has caught at least one fish, he quits. Otherwise, he continues until he catches at least one fish. Let N t denote the Poisson process of the problem. (a) This is the probability of no arrivals in 2 hours. It is given by P ( N 2 = 0) = e - 0 . 5 · 2 = e - 1 . (b) If he catches at least three fish, he must have fished for exactly two hours. Hence, the desired probability is equal to the probability that the number of fish caught in the first two hours is at least three, P ( N 2 3) = 1 - P ( N 2 = 0) - P ( N 2 = 1) - P ( N 2 = 2) = 1 - e - 1 - e - 1 - 1 2 e - 1 = 1 - 5 2 e - 1 . (c) Find the expected number of fish that he catches. Let N be the number of fish the fisherman catches. Let A denote the event that he catches at least one fish in 2 hours. Then, E [ N ] = E [ N | A ] P ( A ) + E [ N | A c ] P ( A c ) = (0 . 5 · 2)(1 - e - 1 ) + 1 · e - 1 = 1 .

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