Let X be a Markov chain and let (nrr≥ 0} be an unbounded increasing sequence of positive integers. Show that Yr Xnr constitutes a (possibly inhomogeneous) Markov chain. Find the transition matrix of Y when nr = 2r and X is: (a) simple random walk, and (b) a branching process. =
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- 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)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.Please show steps to find solution for markov chain
- 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 1There are two printers in the computer lab. Printer i operates for an exponential time withrate λi before breaking down, i = 1, 2. When a printer breaks down, maintenance is called to fix it,and the repair times (for either printer) are exponential with rate μ. (a) Can we analyze this as a birth and death process? Briefly explain your answer.(b) Model this as a continuous time Markov chain (CTMC). Clearly define all the statesand draw the state transition diagram.How to derive this Gauss Markov formulaSuppose 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.(a) Find P(X2 = 1).(b) Find the conditional probability P(X2 = 1|X1 = 1).(c) Find the conditional probability P(X1 = 1|X2 = 1).(d) Find limn→∞ P(Xn = 1).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).Q5. Give an example of a markov chain that is reducible, recurrent and aperiodic.