Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.

A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if
P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j),
where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number
of visits to j before the chain revisits i
i is 1.
Transcribed Image Text:Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.
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