# sol_hw5 - SIMULATION IEOR 4404 SOLUTIONS TO ASSIGNMENT 5 1...

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Unformatted text preview: SIMULATION IEOR 4404 SOLUTIONS TO ASSIGNMENT 5 1. Simulation of non-homogeneous Poisson Process. ( a ) The answer follows from the following property of Poisson processes. Let N 1 ( t ) and N 2 ( t ) be two given independent Poisson processes with rates λ 1 ( t ) and λ 2 ( t ) , respectively. Then the process N ( t ) := N 1 ( t ) + N 2 ( t ) is a Poisson process with rate λ ( t ) = λ 1 ( t )+ λ 2 ( t ) . This property can be easily veri ed by checking if the new process N ( t ) satis es the following: • N (0) = 0 • { N ( t ) , t ≥ } has independent increments • P ( N ( t + h )- N ( t ) = 1) = λ ( t ) h + o ( h ) • P ( N ( t + h )- N ( t ) ≥ 2) = o ( h ) Therefore, the given Poisson process { N ( t ) , t ≥ } with rate λ ( t ) = 3 + 4 1+ t , can be written as the sum of two independent Poisson processes: { N ( t ) , t ≥ } with rate λ ( t ) = 3 and { N 1 ( t ) , t ≥ } with rate λ 1 ( t ) = 1 1+ t . 2. Earthquake model ( a ) Assume we are given a simpli ed version of the earthquake model with intensity λ ( t,x ) . We can view λ ( t,x ) as the joint intensity of time and space. Therefore, if we want to obtain the the characterization of the process only with respect to time (that is,...
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sol_hw5 - SIMULATION IEOR 4404 SOLUTIONS TO ASSIGNMENT 5 1...

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