632-seppalainen-07spex01

632-seppalainen-07spex01 - 632 Introduction to Stochastic...

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Unformatted text preview: 632 Introduction to Stochastic Processes Spring 2007 Midterm Exam 1 Instructions: Show calculations and give concise justifications for full credit. Points add up to 100. In-Class Part 1. Consider the Markov chain on the state space S = { 1 , 2 , 3 , 4 } with transition matrix P = 1 / 3 2 / 3 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 4 1 Rows are indexed from top to bottom and columns from left to write, so p 1 , 1 = 1 / 3, p 2 , 1 = 1 / 2, p 3 , 1 = 1 / 4, etc. (a) (10 pts) Find the transient states, and the closed, irreducible sets of recurrent states. (b) (15 pts) If the chain starts at state 3, what is the probability that it absorbs eventually in state 4? In other words, find the probability P 3 [ there exists a finite number n such that X n = 4 for all n ≥ n ] . (c) (20 pts) Find the limits lim n →∞ p ( n ) 2 , 1 and lim n →∞ p ( n ) 1 , 2 . Explain what theorem you are using and what Markov chain are you applying it to....
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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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632-seppalainen-07spex01 - 632 Introduction to Stochastic...

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