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416-10sol - Denker Spring 2010 416 Stochastic Modeling...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 10 SOLUTIONS: Problem 1: (Problem 40, p. 352) Let N 1 and N 2 be two independent Poisson processes. Show that N ( t ) = N 1 ( t )+ N 2 ( t ) is a Poisson process and determine the rate of N . Solution: We show the properties of the Poisson process with rate λ 1 + λ 2 are satisFed. 1. Since N 1 (0) = N 2 (0) = 0 we also have N (0) = N 1 (0) + N 2 (0) = 0. 2. Let s i < t i , i = 1 , .., n , deFne n disjoint intervals, say t i < s i +1 . Then N ( t i ) - N ( s i ) = N 1 ( t i ) - N 1 ( s i ) + N 2 ( t i ) - N 2 ( s i ) i = 1 , ..., n. Since all random variables { N k ( t i ) - N k ( s i ) : k = 1 , 2; 1 i n } are independent by assumption and the deFnition of a Poisson process, also the random variables N ( t i ) - N ( s i ) are independent. Hence N ( t ) has independent increments. 3. Since N ( t ) - N ( s ) is the sum of the two independent Poisson distributed random variables N k ( t ) - N k ( s ), k = 1 , 2, it is also Poisson distributed with rate being the
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416-10sol - Denker Spring 2010 416 Stochastic Modeling...

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