1. A Markov chain with state space S = {1, 2, 3} has transition matrix P = O HINO 0 1 1 OHINO (a) Draw a directed graph for the Markov chain. (b) Identify all communication classes and classify them as aperiodic or periodic (give the period), and as transient or recurrent. = = f(n), (c) Find the probability of the first return to state i at the nth step, P(T; = n) : for all i = 1, 2, 3 and all n ≥ 1. (d) Use part (c) to calculate the mean recurrent time, µ¿ = E[T], for each state i = 1, 2, 3.

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1. A Markov chain with state space S = {1, 2, 3} has transition matrix
P
-
0 1 0
0
0 1
0
(a) Draw a directed graph for the Markov chain.
(b) Identify all communication classes and classify them as aperiodic or periodic (give
the period), and as transient or recurrent.
=
9
(c) Find the probability of the first return to state i at the nth step, P(T; = n) f(n)
for all i 1, 2, 3 and all n ≥ 1.
=
(d) Use part (c) to calculate the mean recurrent time, µ = E[T], for each state
i = 1, 2, 3.
Transcribed Image Text:1. A Markov chain with state space S = {1, 2, 3} has transition matrix P - 0 1 0 0 0 1 0 (a) Draw a directed graph for the Markov chain. (b) Identify all communication classes and classify them as aperiodic or periodic (give the period), and as transient or recurrent. = 9 (c) Find the probability of the first return to state i at the nth step, P(T; = n) f(n) for all i 1, 2, 3 and all n ≥ 1. = (d) Use part (c) to calculate the mean recurrent time, µ = E[T], for each state i = 1, 2, 3.
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