Consider the random walk on the integers {0, 1, 2, 3} which takes steps +1 (to t

Question

Consider the random walk on the integers {0, 1, 2, 3} which takes steps +1 (to the right) with probability 1 reflection; this means that a step from 1 to 0 is always followed by a step from 0 to 1, and a step from 2 to 3 is always followed by a step from 3 to 2. and 1 (to the left) with probability 2 3 , except at the endpoints where there is 3 to 2

(a) Determine the transition matrix for this Markov chain.

(b) Suppose the Markov chain has been running for a long time. What fraction of time has it

spent in state 0?

Explanation / Answer

The transistion matrix is

The state {0} is closed communicating class and its fraction of time is 4(2/3)=8/3=2.667.

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