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Unformatted text preview: 632 Introduction to Stochastic Processes Spring 2004 Final Exam Instructions: Show calculations and give concise justifications for full credit. Points add up to 200. 1. Four cages labeled A , B , C and D are connected by tubes as indicated in the diagram below. A rat runs from one cage to the other. Whenever it leaves a cage, it chooses at random among the tubes leading out of the cage. (So for example when the rat leaves A , it chooses one of B , C and D with equal probability, and when it leaves B , it chooses one of A and C with equal probability.) Let X n denote the n th cage the rat visits. Suppose the rat starts in A , so X = A . (a) (30 pts) What is the expected number of cages the rat visits until its first return to cage A , not counting the first and last visits to A ? (b) (25 pts) Find lim n →∞ P { X n = A, X n +1 = B, X n +2 = A } in other words the longterm limiting probability that we see the rat visit A B A ....
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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.
 Spring '07
 Seppalainen

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