. A Markov chain (Xn, n = 0, 1, . . . ) has state space S = {1, 2, 3, 4} and transition matrixP =0 2/5 3/5 01/4 0 1/2 1/40 3/4 0 1/40 0 0 1(a) Draw the transition diagram for this Markov chain. [2 marks](b) Find P (X3 = 1 | X0 = 1, X1 = 2, X2 = 1). [1 mark](c) Find P (X1 = 2, X2 = 1, X3 = 3, X4 = 4 | X0 = 3).
Question
. A Markov chain (Xn, n = 0, 1, . . . ) has state space S = {1, 2, 3, 4} and transition matrixP =0 2/5 3/5 01/4 0 1/2 1/40 3/4 0 1/40 0 0 1(a) Draw the transition diagram for this Markov chain. 2 marks Find P (X3 = 1 | X0 = 1, X1 = 2, X2 = 1). 1 mark Find P (X1 = 2, X2 = 1, X3 = 3, X4 = 4 | X0 = 3).
Solution
(a) The transition diagram for this Markov chain would look like this:
1 -> 2 (with probability 2/5) 1 -> 3 (with probability 3/5)
2 -> 1 (with probability 1/4) 2 -> 3 (with probability 1/2) 2 -> 4 (with probability 1/4)
3 -> 2 (with probability 3/4) 3 -> 4 (with probability 1/4)
4 -> 4 (with probability 1)
(b) The probability P(X3 = 1 | X0 = 1, X1 = 2, X2 = 1) is not dependent on X0 and X1 due to the Markov property (the future is independent of the past given the present). Therefore, we only need to consider P(X3 = 1 | X2 = 1), which is the probability of going from state 1 to state 1 in one step. Looking at the transition matrix, we see that there is no direct transition from state 1 to state 1. Therefore, P(X3 = 1 | X2 = 1) = 0.
(c) The probability P(X1 = 2, X2 = 1, X3 = 3, X4 = 4 | X0 = 3) can be calculated by multiplying the probabilities of each transition:
P(X1 = 2 | X0 = 3) * P(X2 = 1 | X1 = 2) * P(X3 = 3 | X2 = 1) * P(X4 = 4 | X3 = 3)
From the transition matrix, we find these probabilities to be:
P(X1 = 2 | X0 = 3) = 3/4 P(X2 = 1 | X1 = 2) = 1/4 P(X3 = 3 | X2 = 1) = 3/5 P(X4 = 4 | X3 = 3) = 1/4
Multiplying these together gives:
(3/4) * (1/4) * (3/5) * (1/4) = 9/320.
Similar Questions
Quiz question For each of the following transition matrices for Markov chains with state spaceS = {1, 2}, write down the Full Balance Equations and find the equilibrium distribution(s).(a) P =0.4 0.60.8 0.2(b) P =0.82 0.180.82 0.18
Quiz Question Consider the Markov chain with transition diagram345210.10.50.50.60.30.50.20.40.30.50.50.20.4i.e., with transition matrixP =0.6 0.4 0 0 00.5 0.5 0 0 00 0.5 0 0.5 00 0.5 0.3 0 0.20 0.1 0.2 0.4 0.3 .(a) Starting from state 5, what is the most likely two-step path? What is the probability(starting from state 5) of following that two-step path?(b) In this Markov chain, can you get from every state to every other state eventually?
If the initial state probability distribution of Markov chain is ( ) ( )and transition probability matrix of the chain is ( ). Compute theprobability distribution of the chain after 2 steps
Given a transition matrix [[0.6, 0.3, 0.1], [0.2, 0.5, 0.3], [0.2, 0.4, 0.4]] for states [Person, Organization, Location], calculate the probability of transitioning from 'Person' to 'Organization' and then to 'Location'.Question 9Answera.0.09b.0.3c.0.6d.0.06
A particle moves in a circle through points that have been marked 0, 1, 2, 3, 4 in a clockwise direction.order. The particle starts at point 0. At each step it has probability 0.35 of moving one point clockwise (0follows 4) and 0.65 of moving one point counter clockwise.Let Xn denote its location on the circle after step n.a). Construct the one step transition matrix for Xn.b) . Determine n-step transition matrix for n =5.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.