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# 03fafin - 632 Introduction to Stochastic Processes Fall...

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632 Introduction to Stochastic Processes Fall 2003 Final Exam Instructions: Justify non-obvious statements for full credit. Quote re- sults from class accurately. Points add up to 200. 1. Consider the discrete-time Markov chain with state space S = Z + = { 0 , 1 , 2 , 3 ,... } and transitions p (0 , 0) = q 0 , p (0 , 1) = p 0 , and p ( n,n - 1) = q n , p ( n,n + 1) = p n for all n 1, where { q k ,p k : k 0 } are given positive numbers that satisfy p k + q k = 1 for each k . (a) (20 pts) Find a condition on the parameters that is equivalent to the existence of an invariant distribution π for the model, and give a formula for π . Does there exist an invariant distribution for this chain that is not reversible? (b) (20 pts) Let T 2 = min { n 1 : X n = 2 } be the time of the ﬁrst visit to state 2 after time 0. Find E 0 [ T 2 ], the expected time to reach state 2 starting from state 0. (c) (20 pts) Let T 2 1 be the time of the second visit to state 1 after time 0. Find E 0 [ T 2 1 ]. (In parts (b) and (c), give answers for all values of the

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## This note was uploaded on 08/11/2008 for the course MATH 632 taught by Professor Seppalainen during the Spring '07 term at Wisconsin.

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03fafin - 632 Introduction to Stochastic Processes Fall...

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