fin.s06 - STAT 150 FINAL EXAM, Spring 2006, JWP...

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STAT 150 FINAL EXAM, Spring 2006, JWP Lastname,Firstname and SID#: 1. Suppose a Markov chain ( X n ,n = 0 , 1 ,... ) has transition matrix P . Let i,j,k be states of the chain, and N a positive integer. Derive expressions involving summations and powers of P for the following quantities: a) Given X 0 = i , the expected number of times n with 1 n N and X n = j . b) Given X 0 = i , the expected number of times n with 1 n N and X n = j and X n +1 = k . c) Given X 0 = i , the probability that T i > 2, where T i is the least n 1 such that X n = i , and T i = if there is no such n . 2. Suppose that ( X 1 ( t ) ,t 0) and ( X 2 ( t ) ,t 0) are two independent Markov chains with continuous time parameter, each with state space { 0 , 1 } , with transition rates λ from 0 to 1 and μ from 1 to 0, for i = 1 , 2. Let S 2 ( t ) := X 1 ( t ) + X 2 ( t ). a) Explain why ( S 2 ( t ) ,t 0) is a Markov chain, and describe its transition rate matrix. b) What is the limit distribution of S 2 ( t ) as t → ∞ ? c) Generalize part b) to
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This note was uploaded on 10/02/2009 for the course UGBA 08547 taught by Professor Odean during the Spring '09 term at Berkeley.

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fin.s06 - STAT 150 FINAL EXAM, Spring 2006, JWP...

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