1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose...

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Question

1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} b

math Advanced-Math

Answer 1

Answer #1

$X_n$ is a discrete times Markov chain with $p_{ii} >0$ . We have to show that $T minn 1 Xni}$ is a geometric random variable.

Now $P(T k) P(X =i for <k, Xk i)$

$P(Xi, X = i, . .. , Xk-1 = i, X i)$

$=P(XkiXk-1 )P(X-1 iXk_2 = i)... P(X1 Xi)$ By Markov property

$= (1 -Pii)P. .. Pi = (1 - Pii)P 22$

$P(T k) (1 - Pi P$ .

This is the distribution of a geometric random variable with win probability $Pi$ .

Answer 2

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