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Unformatted text preview: IEOR 4701: HMWK 4 1. You arrive at the West 96th Street Subway station to go Downtown. Suppose Express trains arrive according to a Poisson process at rate 4 (per hour), and independent of this, Local trains arrive according to a Poisson process at rate 7 (per hour). You decide to take the first train (whatever it is) that arrives. (a) What is the expected number of express trains to arrive during the 2 hour interval 2 4PM? SOLUTION: E ( N 1 (4) N 1 (2)) = E ( N 1 (2)) = λ 1 2 = 4(2) = 8. (Stationary increments.) (b) What is the variance of the total number of trains (regardless of type) to arrive during the 2 hour interval 2 4PM? SOLUTION: The superposition is a Poisson process at rate λ = λ 1 + λ 2 = 11. V ar ( N (2)) = E ( N (2)) = 2 λ = 22. (c) What is the probability that you take an Express train? SOLUTION: λ 1 / ( λ 1 + λ 2 ) = 4 / 11. (d) What is the expected amount of time you wait for a train to arrive? SOLUTION: Z = min { X,Y } where X ∼ exp ( λ 1 ) (first arrival time of an express) and (indepen dently) Y ∼ exp ( λ 2 ) (first arrival time of a local). Thus Z ∼ exp ( λ ); E ( Z ) = 1 /λ = 1 / 11. (e) Given that you took an Express train (it arrived first), what was the expected amount of time you waited for it to arrive? SOLUTION: E ( Z  X < Y ) = E ( Z ) = 1 / 11. ( Z is independent of which of the two is the minimum.) (f) What is the probability that you must wait longer than 12 minutes? SOLUTION: 12 minutes is 1 / 5 an hour. P ( N (12) = 0) = e λ (1 / 5) = e 11 / 5 . (g) What is the probability that exactly 3 express trains arrive before the first local?...
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 Spring '10
 KarlSigma
 Poisson Distribution, Probability theory, Exponential distribution, Poisson process, Markov chain

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