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February 12th, 2012, 05:31 AM   #1
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Markov chain

A boy and a girl move to the same two-bar-town on the same day. Each night, boy visits one bar, starting the first night with bar 1 and continues by selecting a bar for the next night according to a Markov chain with transition matrix P below. Similarly, girl visits one bar a night,starting with bar 2 and selecting the next bar according to Q below:



Hint: The progress of boy and girl finding each other can be modelled as a single Markov chain where only one (!) state is absorbing.
a) Find the probability that boy visits bar 1 and girl visits bar 2 on the nth night.
b) Let N denote the number of the night when girl meets boy. Compute the expected number of nights E[N] it takes for boy and girl to meet. Hint: Either compute the distributionof N, or develop an invariance equation for E[N].
c) Find the probability that they meet in bar 1.
d) Find the distribution of the time of their meeting (distribution of N).

Thank you very much for your help in advance!
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February 12th, 2012, 04:20 PM   #2
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Re: Markov chain

Do first a eigensystemdecomposition for P and Q, then it is easier to calculate P^n and Q^n.
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