# The probability a call started at s has ended by time

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Unformatted text preview: ars that have arrived by time t with exactly j people. The somewhat remarkable fact is: Theorem 2.11. Nj (t) are independent Poisson processes with rate P (Yi = j ). Why is this remarkable? There are two “surprises” here: the resulting processes are Poisson and they are independent. To drive the point home consider a Poisson process with rate 10 per hour, and then ﬂip coins to determine whether the arriving customers are male or female. One might think that seeing 40 men arrive in one hour would be indicative of a large volume of business and hence a larger than normal number of women, but Theorem 2.11 tells us that the number of men and the number of women that arrive per hour are independent. Proof. To begin we suppose that P (Yi = 1) = p and P (Yi = 2) = 1 p, so there are only two Poisson processes to consider: N1 (t) and N2 (t). We will check the second deﬁnition given in Theorem 2.7. It should be clear that the independent increments property of the Poisson process implies that the pairs of increments (N1 (ti ) N1 (ti 1 ), N2 (ti ) N2 (ti 1 )), 1in are independent of each other. Since N1 (0) = N2 (0...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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