. Consider a Markov Chain with state space {0,1, 2, 3, 4} and transition matrix 2 3 4 1 0 0 1 1/3 1/3 1/3 .0 1/3 1/3 1/3 0 0 1/3 1/3 1/3 0 - 0 0 1 P = 2 3 4 (n) Then limn-o P23 equals
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- Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).A Markov chain {x}=0,1,2,.. satisfies the difference equation X = Axk-1 for every k ≥ 1, where - (0.9 A = {) 0.8 0.6 0.2 0.4 (i) Find the general term x for k≥ 1. (ii) What happens to x as k→ ∞o? - (0.3). and Xo =2.- Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6, (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems in the notes. (b) Find the limiting distribution for this Markov chain. (100) (c) Without doing any more calculations, what can you say about p₁,1 (100) ? and P2,1
- Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1Suppose that a Markov chain has transition probability matrix 1 2 1 P (1/2 1/2 2 1/4 3/4 (a) What is the long-run proportion of time that the chain is in state i, i = 1,2 ? 5. What should r2 be if it is desired to have the long-run average (b) Suppose that ri reward per unit time equal to 9?2. Let Xo, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix (1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6/ (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it.--.*/ 'VOLC. .. A state vector X for a three-state Markov chain is such that the system is as likely to be in state 2 as in state 3 and is five times as likely to be in state 1 as in 2. Find the state vector X.Consider a Markov chain {X, : n = 0, 1, - .-} on the state space S = {1,2,3,4} with the following transition matrix: 1/3 2/3 1/2 1/2 P = 1/4 3/4 1/4 1/4 1/2 Find Pr(X7 = 2|X1 = 3). %3D Determine the class(es) of the above Markov chain. Specify which state is recurrent and which state is transient. Justify your results.Suppose the transition matrix for a Markov Chain is T = stable population, i.e. an x0₂ such that Tx = x. ساله داد Find a non-zeroGiven transition matrix P8- 2 and [V]g [V]B = 1 3 a. none of these b. =DPg' B[V]g =| %3D OC = PB- B'[V]e" !! d. = Pg - B[V]g = 11 = Pg' - B[V]g• =A Markov Chain has the transition matrix r-[% *]. P = and currently has state vector % % ]: What is the probability it will be in state 1 after two more stages (observations) of the process? (A) % (B) 0 (C) /2 (D) 24 (E) 12 (F) ¼ (G) 1 (H) 2242. Let Xo, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7] 0 3/7 3/7 1/3 1/3 1/3 0 2/3 1/6 1/6, 0 (b) Find the limiting distribution for this Markov chain. (100) (100) ? Without doing any more calculations, what can you say about P₁,1 and P2,1SEE MORE QUESTIONS