18445_pr1sol

# 18445_pr1sol - Stochastic Processes – 18.445 MIT fall...

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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 ¯ π such that lim n →∞ πP n = ¯ π for every probability distribution π . (b) lim n →∞ P n = ¯ π . . . ¯ π , for some probability distribution ¯ π . (c) There exists a probability distribution ¯ π such that ¯ πP = ¯ π . (d) 1 is an eigenvalue of P with multiplicity 1, and all other eigenvalues λ have | λ | < 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. α i with prob. β i + 1 with prob.with prob....
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18445_pr1sol - Stochastic Processes – 18.445 MIT fall...

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