9. Let i be a transient state of a continuous-time Markov chain X with X (0) : total time spent in state i has an exponential distribution. = i. Show that th
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.There 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.Suppose 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?
- Let X, be the Markov chain with state space Z and transition probability Pa,z+1 = P, Pa,2-1 = 1- p, where p > 1/2. Assume X, = 0. (a) Let Y = min{Xo, X1,...}. What is the distribution of Y? (b) For positive integer k, let T = min{n : X, = k} and let e(k) = E[TR]. Explain why e(k) = ke(1). (c) Find e(1). Hint: part (b) might be helpful. (d) Use (c) to give another proof that e(1) = o if p = 1/2.Let (X0, X1, X2, . . .) be the discrete-time, homogeneous Markov chain on state space S = {1, 2, 3, 4, 5} with X0 = 3 and transition matrixLet P = (0.9 0.9 0.1 0.2 0.8 with two states A and B. be a transition matrix for a Markov Chain 1. What proportion of the state A population will be in state B after two steps Number 2. What proportion of the state B population will be in state B after two steps Number 3. Find the steady state vector x x1= Number X2= Number Write the results accurate to the 3rd decimal place
- A Continuous-Time Markov Chain Consisting of Two States. Consider a machine that works for an exponential amount of time having mean 1/A before breaking down; and suppose that it takes an exponential amount of time with mean 1/µ to repair the machine. If the machine is in the operational condition at time 0, what is the probability that it will be working at time t = 5 ? HINT: write down the backward Kolmogorov differential equations and solve it.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.(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).Let X be a random variable with sample space {1,2, 3} and probability distribu- (G 1 ). Find a transition matrix P such that the Markov chain {X„} tion T = simulates X.
- 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.= 7. Let X be a continuous-time Markov chain with transition probabilities pij (t) and define F; = inf{t > T₁: X(t) =i} where T₁ is the time of the first jump of X. Show that, if gii #0, then P(F; <∞ | X (0) = i) = 1 if and only if i is persistent.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 1SEE MORE QUESTIONS
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