In Exercises 1–6, consider a Markov chain with state space with {1, 2, …, n} and the given transition matrix. Find the communication classes for each Markov chain, and state whether the Markov chain is reducible or irreducible.
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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_forward12. 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.)arrow_forwardConsider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show that the steady state matrix X depends on the initial state matrix X0 by finding X for each X0. X0=[0.250.250.250.25] b X0=[0.250.250.400.10] Example 7 Finding Steady State Matrices of Absorbing Markov Chains Find the steady state matrix X of each absorbing Markov chain with matrix of transition probabilities P. b.P=[0.500.200.210.300.100.400.200.11]arrow_forward
- Please help me with this question attached below. Thank you.arrow_forwardSuppose that a Markov chain with 3 states and with transition matrix P is in state 3 on the first observation. Which of the following expressions represents the probability that it will be in state 1 on the third observation? (A) the (3, 1) entry of P3 (B) the (1,3) entry of P3 (C) the (3, 1) entry of Pª (D) the (1,3) entry of P2 (E) the (3, 1) entry of P (F) the (1,3) entry of P4 (G) the (3, 1) entry of P2 (H) the (1,3) entry of Parrow_forwardA Markov chain model for a species has four states: State 0 (Lower Risk), State 1 (Vulnerable), State 2 (Threatened), and State 3 (Extinct). For t 2 0, you are given that: 01 zit = 0.03 12 t= 0.05 23 Hit = 0.06 This species is currently in state 0. Calculate the probability this species will be in state 2 ten years later. Assume that reentry is not possible. (Note: This question is similar to #46.2 but with constant forces of mortality) Possīble Answers A 0.02 0.03 0.04 D 0.05 E 0.06arrow_forward
- Determine whether the statement below is true or false. Justify the answer. If (x) is a Markov chain, then X₁+1 must depend only on the transition matrix and xn- Choose the correct answer below. O A. The statement is false because x, depends on X₁+1 and the transition matrix. B. The statement is true because it is part of the definition of a Markov chain. C. The statement is false because X₁ +1 can also depend on X-1 D. The statement is false because X₁ + 1 can also depend on any previous entry in the chain.arrow_forwardan 15 A continuous-time Markov chain (CTMC) has three states (1, 2, 3). ed dout of The average time the process stays in states 1, 2, and 3 are 3, 11.9, and 3.5 seconds, respectively. question The steady-state probability that this CTMC is in the second state ( , ) isarrow_forwardLet (Xn)nzo be a Markov chain with state space S = {A, B, C} and transition matrix P given by Compute PBB, n € N. P= 14 12 L 1 4 D| D|- NIT AT A L 1 2 4 4 4 1 2arrow_forward
- Let {Xn, n = 0, 1, 2, . . .} be a three-state Markov chain with S = {0, 1, 2} and the transition probability matrix [0.1 0.3 0.6] P = 0.7 0.3 0 [0.5 0 0.5 State O represents an operating state of some system, while states 1 and 2 represent repair states (corresponding to two types of failures). We assume that the process begins in state ✗( = 0, and then the successive returns to state 0 from the repair state form a renewal process. Determine the mean duration of one of these renewal intervals. E[renewal interval] = =arrow_forward1. Identify all absorbing states in the Markov chains having the following matrix. Decide whether the Markov chain is absorbing. 1 2 3 1[ 1 a) 2 0.3 0.5 0.2 3 1. 1 4 [0.6 0 0.4 01 1 a) 2 0.9 0.1 0 0 0 4 2. Find the first three powers of each of the transition matrix. For each transition matrix, find the probability that state 1 changes to state 2 after three repetition of the experiment. a) C = l0.72 0.28 0.5 [0.8 0.1 0.1] b) E = |0.3 0.6 0.1 1arrow_forwardPlease describe the steps you used to get the solution to the problem provided in the image below.arrow_forward
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