18445_pr1sol

18445_pr1sol - Stochastic Processes 18.445 MIT, fall 2011...

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Unformatted text preview: Stochastic Processes 18.445 MIT, fall 2011 Practice Mid Term Exam 1 October 25, 2011 Problem 1: . Let X 1 ,X 2 ,X 3 ,... be a Markov chain on a finite state space S = { 1 ,...,N } with transition matrix P . Among the following statements, say which implies which. (a) There exists a probability distribution such that lim n P n = for every probability distribution . (b) lim n P n = . . . , for some probability distribution . (c) There exists a probability distribution such that P = . (d) 1 is an eigenvalue of P with multiplicity 1, and all other eigenvalues have | | < 1. (e) P ij > 0 for all i,j S . (f) There exists n > 0 such that P n ij > 0 for all i,j S (g) P n ij > 0 for all i,j S and n > 0. (h) For all i,j S there exists n > 0 such that P n ij > 0. (i) X 1 ,X 2 ,X 3 ,... is an irreducible Markov chain. (j) X 1 ,X 2 ,X 3 ,... is an irreducible aperiodic Markov chain. Solution: ( e ) ( g ) = ( f ) ( j ) = ( d ) = ( a ) ( b ) ( h ) ( i ) = ( c ) Problem 2: . Let X 1 ,X 2 ,X 3 ,... be a Markov chain on Z such that X = 0 and, conditioned on X n = i , we have X n +1 = i- 1 with prob. i with prob. i + 1 with prob.with prob....
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18445_pr1sol - Stochastic Processes 18.445 MIT, fall 2011...

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