Homework 1 2 3 - POLITECNICO DI TORINO STOCHASTIC PROCESSES...

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Unformatted text preview: POLITECNICO DI TORINO STOCHASTIC PROCESSES 2008-2009 HOMEWORK 6: DUE TO OCT 27,2008 Ref . S. Ross, Introduction to Probability models-Eight edition, Chapter 5. Exercises In order to solve the following exercises, the following proposition is needed. Proposition 0.1. Consider a Poisson process { N ( t ) ,t ≥ } having rate λ , and suppose that each time and event occurs it is classified as a type I event type with probability p or a type II event with probability 1- p , independently of all other events. Let N 1 ( t ) and N 2 ( t ) denote respectively the number of type I and type II events occurring in [0 ,t ] . Note that N ( t ) = N 1 ( t ) + N 2 ( t ) . Then (i) { N 1 ( t ) ,t ≥ } and { N 2 ( t ) ,t ≥ } are both Poisson processes having respec- tively rates λp and λ (1- p ) . (ii) Case ( i ) extends to the case of M types, with N ( t ) = N 1 ( t ) + ... + N M ( t ) , where { N j ( t ) } is a Poisson process of rate λp j , with ∑ p j = 1 . (iii) Suppose that the probability that an event is classified as a type i event, i = 1 ,...,k , depends on the time the event occurs. Specifically, suppose, depends on the time the event occurs....
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Homework 1 2 3 - POLITECNICO DI TORINO STOCHASTIC PROCESSES...

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