# ass5 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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STAT455/855 Fall 2009 Applied Stochastic Processes Assignment #5 Due Friday, Dec.4 1. Ross, Chapter 6, #1 (8th Edition: Chapter 6, #1). 2. PASTA: Poisson Arrivals See Time Averages . Consider a continuous time Markov chain observed at the times of a Poisson process with rate λ . Let X = { X ( t ) : t 0 } be a continuous time Markov chain with stationary distribution π . Let S 1 , S 2 , . . . be the event times of a Poisson process with rate λ . Deﬁne Y n = X ( S n ) for n 1. Then Y = { Y n : n 1 } is a discrete time Markov chain. Show that the stationary distribution of Y is also π . Hint : Show that the transition probabilities for the discrete time chain Y are given by q ij = Z 0 p ij ( t ) λe - λt dt, where p ij ( t ) is the ( i, j )th transition probability function for the X chain. 3. Let X = { X ( t ) : t 0 } be an irreducible continuous time Markov chain with state space S , inﬁnitesimal generator G = (( g ij )) i,j S and parameters v i , p ij and q ij = v

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## This note was uploaded on 09/15/2010 for the course STAT 455/855 taught by Professor Glentakahara during the Fall '09 term at Queens University.

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ass5 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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