Question

Let Xn be a Markov Chain with state space S = 0, 1, 2, · · ·. For each of the following transition probabilities,state if the chain is positive recurrent,null recurrent,or transient. If it is positive recurrent, give the stationary probability distribution : (a) Pi0 = 1 i+3 , Pi(i+1) = i+2 i+3 (∀i ∈ S) (b) Pi0 = i+2 i+3 , Pi(i+1) = 1 i+3 (∀i ∈ S) (c) Pi0 = 1 i 2+3 , Pi(i+1) = i 2+2 i 2+3 (∀i ∈ S) (d) Assume that we keep rolling on a fair dice. What is the expected number of rolls before we see k consecutive 5s.

          Let Xn be a Markov Chain with state space S = 0, 1, 2, · · ·. For each of
the following transition probabilities,state if the chain is positive recurrent,null recurrent,or transient. If it is positive recurrent, give the stationary probability distribution
:
(a) Pi0 =
1
i+3 , Pi(i+1) =
i+2
i+3 (∀i ∈ S)
(b) Pi0 =
i+2
i+3 , Pi(i+1) =
1
i+3 (∀i ∈ S)
(c) Pi0 =
1
i
2+3 , Pi(i+1) =
i
2+2
i
2+3 (∀i ∈ S)
(d) Assume that we keep rolling on a fair dice. What is the expected number of rolls
before we see k consecutive 5s.

Added by Bhushan M.

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