02faex2 - 632 Introduction to Stochastic Processes Fall...

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632 Introduction to Stochastic Processes Fall 2002 Midterm Exam II Instructions: Hand in problem 1 for 50 points, problem 2 for 30 points, and one other problem for 20 points. Show calculations and justify non- obvious statements for full credit. 1. Fix a constant α (0 , 1), and consider again the Markov chain on the state space S = { 1 , 2 , 3 , 4 , 5 , 6 } with transition matrix P = 0 0 1 - α α 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 - α 0 α 0 In Exam 1 we found the invariant distribution π = ± 1 3 - α , 0 , 1 - α 3 - α , 1 3 - α , 0 , 0 ² . (a) State the hypotheses under which we proved the convergence theorem p n ( x,y ) π ( y ) for Markov chains. Find lim n →∞ p n (1 , 3) and explain briefly how the hypotheses of the theorem are met. (b) Find the limiting probability lim n →∞ P 3 [ X n = 3 ,X n +3 = 3 ,X n +6 = 3].
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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 University of Wisconsin.

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02faex2 - 632 Introduction to Stochastic Processes Fall...

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