The following Markov chain is irreducible 1/3 0 0 0 1 1/5 2/3 0 4/5
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.Can a Markov chain in general have an infinite number of states? O yes no Previous Sin5 Consider the Markov chain with transition matrix 1/2 (1/4 3/4) P = Find the fundamental matrix Z for this chain. Compute the mean first passage matrix using Z.
- Q5. Give an example of a markov chain that is reducible, recurrent and aperiodic.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).How many DAGS are Markov equivalent to the “chain DAG" X1 → X2 → · .. → X, (excluding itself)?
- 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).Suppose you toss a six-sided die repeatedly until the product of the last two outcomes is equal to 12. What is the average number of times you toss your die? Construct a Markov chain and solve the problem.Consider two machines that are maintained by a single repairman. Each machine functions for an exponentially distributed amount of time with rate λ=2 before it fails. The repair times for each unit are exponential with rate μ=1.The Markov chain model for this situation has state space indicating the number of machines that are in the repair shop: S={0,1,2}. Notice that you move from 0 to 1 if one of the two machines breaks down. You move from 1 to 0 when the machine in the repair room is repaired.