2. Consider a Markov chain X=(Xn)n≥0 with state space S ={1, 2, 3} and transition matrix[0.5 0 0.5]0 100.4 0 0.6(a) Which states are transient and which are recurrent?(b) Is this markov chain irreducible or reducible?

First, let's recall what transient and recurrent states are:Transient state: A state is called…

Answered: 2. Consider a Markov chain X = (Xn)n20…

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
The sequence (Xn)n>o is a Markov chain with transition matrix 0 을 0 0 0 3 1 0 0 0 4 4 0 } 0 0 0 1 2 1 1 0 0 3 3 0 } 0 } 0 0 0 0 0 0 0 (a) Draw the transition diagram for (Xn). (b) Determine the communicating classes of the chain, stating whether each class is recurrent or transient. (c) Determine the period of each communicating class.
a)
Recall that the graph of a discrete-time, finite-state, homogeneous Markov chain on two states 1, 2 with transition matrix is CO (: :) 1 T = If instead a new Markov chain on three states {1, 2, 3} has transition matrix 10 1 0 (a) Draw the corresponding graph for this new Markov chain. (b) Is the Markov chain with transition matrix T irreducible?
2. Consider a Markov chain {Xn}n≥o having the following transition diagram: 1/2 1 2 5 1/4 3 4 1/2 1/4 6 7 1/2 For this chain, there are two recurrent classes R₁ = {6,7} and R₂ = {1,2,5}, and one transient class R3 = {3,4}. (a) Find the period of state 3. (b) Find f33 and f22. (c) Starting at state 3, find the probability that the chain is absorbed into R₁. (d) Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R₁ or R₂. Note: there are missing transition probabilities for this chain, but no impact for your solution.
C)

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2. Consider a Markov chain X = (Xn)n20 with state space S = {1, 2, 3} and transition matrix [0.5 0 0.5] 0 1 0 0.4 0 0.6 (a) Which states are transient and which are recurrent? (b) Is this markov chain irreducible or reducible?