Follow the plan of Exercise 25 to confirm Theorem 5 for the Markov chain with transition matrix
where 0 < p < 1
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Linear Algebra and Its Applications (5th Edition)
- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.arrow_forwardIf she made the last free throw, then her probability of making the next one is 0.7. On the other hand, If she missed the last free throw, then her probability of making the next one is 0.2. Assume that state 1 is Makes the Free Throw and that state 2 is Misses the Free Throw. (1) Find the transition matrix for this Markov process. P =arrow_forwardLet (X0, X1, X2, . . .) be the discrete-time, homogeneous Markov chain on state space S = {1, 2, 3, 4, 5} with X0 = 3 and transition matrixarrow_forward
- 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 2/3 1/6 1/6, 0 0 (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. (c) Without doing any more calculations, what can you say about p1100) and p(100)?arrow_forward2. Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 0 1/2 1/2 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 Theoremsarrow_forwardfourth digit is 2arrow_forward
- please solve on paperarrow_forwardSuppose 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).arrow_forwardA Markov chain X₁, X₁, X₂ ... on the states 0, 1, 2 has the transition probability matrix 1 2 0.2 0.7 P 10.2 0.2 0.6 2 0.6 0.1 0.3 and initial distribution Po = P(X₁ = 0) = 0.2, P₁ = P(Xo = 1) = 0.3, and P2 = P(Xo = 2) = 0.5. (1) Compute the two-step transition matrix; (2) What is P(X3 = 1|X₁ = 0)? (3) What is P(X3 = 0, X5 = 2|X₂ = 1)? (4) What is P(Xo = 2, X₂ = 0, X3 = 1)? 0 0 0.1arrow_forward
- 1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. • If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine…arrow_forwardA Markov chain X₁, X₁, X₂... on states 0, 1, 2 has the transition probability matrix P (shown on the right) and initial distribution P(X₁ = 0) = 0.3, P(X。 = 1) = 0.4 and P(Xo 2) = 0.3. (1) Show that P is regular. (2) Find P(Xo = 2, X₁ = 0, X₂ = 1). (3) Find P(X₂ = 2, X₁ = 1|X₁ = 0). (4) Determine the limiting distribution. = 0 0 0.7 P = 1 0 20.5 1 2 0.2 0.1|| 0.6 0.4 0 0.5arrow_forwardSuppose 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?arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning