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08hw2 - N = inf n ≥ 1 X n ≥ c mark the ﬁrst...

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632 Introduction to Stochastic Processes Fall 2008 Part of Homework 2 1. You are trying to cross a busy highway. You need time c to cross safely. Interarrival times between cars are i.i.d. with common distribution p k = pq k - 1 for k N . Find the expected time of your arrival on the other side of the road. A little more precisely: You arrive at the side of the road at time 0. Let S 0 = 0. The arrival times of cars are S 1 , S 2 , S 3 ,... where S n = X 1 + ··· + X n and the random variables { X i } are i.i.d. with P ( X i = k ) = pq k - 1 for positive integers k . Assume 0 < p < 1, q = 1 - p , and let c be a positive integer. Let
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Unformatted text preview: N = inf { n ≥ 1 : X n ≥ c } mark the ﬁrst interarrival time of length at least c . As soon as the previous car has passed you cross the road and arrive to the other side at time T = S N-1 + c . Find E ( T ) in terms of the two parameters of the model, p and c . If you use Wald’s identity, be sure to check its hypotheses. After you found an answer, try to check that it makes sense. For example if c = 1 the answer should be pretty obvious. Does your formula give the right answer?...
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