solve part c 4.Simple Markov ChainLet X0, X1, X2,... be a Markov chain with state space S={1, 2, 3}, transition matrix0.1 0.1 0.8P =0.2 0.4 0.40.3 0.7 0and initial distribution aT=(0.1, 0.6, 0.3). Compute(a) P[X7 = 1|X6 = 3],(b) P[X9 = 3|X₁ = 1, X5 = 2, X7 == 1],(c) P[X0 = 2|X₁ = 3],(d) E[X2].

Given that: P=0.1 & 0.1 & 0.8 \\ 0.2 & 0.4 & 0.4 \\ 0.3 & 0.7 & 0 and we…

Answered: 4. Simple Markov Chain Let X0, X1,…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 12EQ: 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction...

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Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.
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.)
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.
2.2 Let Xo. X1,... be a Markov chain with transition matrix 1 2 3 1 1/2 1/2) 1 3(1/3 1/3 1/3) and initial distribution a = (1/2,0, 1/2). Find the following: (a) P(X, = 1|X¡ = 3) (b) P(X1 = 3, X2 = 1) (c) P(X1 = 3|X2 = 1) (d) P(X9 = 1|X1 = 3, X4 = 1,X7 = 2) %3D
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)?
2. 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 Theorems
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?
4. Suppose Xn is a two-state Markov chain whose transition probability matrix P is 0 1 0 Q 1-a 1 1-8 B Then, Zn (Xn-1, Xn) is a Markov chain having the four states (0, 0), (0, 1), (1,0), and (1,1). Determine the transition probability matrix.
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 2/3 1/6 1/6, 0 (b) Find the limiting distribution for this Markov chain. (100) (100) ? Without doing any more calculations, what can you say about P₁,1 and P2,1
A 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 0.1
1.1. A Markov chain X,, X, X,, ... has the transition probability matrix 1 2 0 ||0.7 0.2 0.1 0.6 0.4 0 0.5 P = 1 2||0.5 Determine the limiting distribution.
Suppose Xn, n = 0, 1, ... is a two-state Markov chain with states Sx = {0, 1}, whose transition probability matrix is 0.1 0.9 0.95 0.05 Then, Zn = (Xn-1, Xn), n = = 1,2,. That is, Zn i j iff Xn-1 i and Xn Pz= = = Px = is a four-state Markov chain with states Sz = {00, 01, 10, 11}. = j. Determine its transition probability matrix.

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