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Consider a two-state Markov chain with transition matrix T = [1/3 2/3; 3/4 1/4] Assume the chain starts in state 1 at step n = 0. (a) What is the probability that the chain is in state 1 at step n = 3? (b) Compute the eigenvalues and eigenvectors of T (do this by hand, show all work). (c) Use your answers to (b) to diagonalize T, and use the diagonalization to determine the behavior of Tn as n → ∞.
Madhur L.
A Markov chain X0, X1, X2, .. with states 1, 2, 3, has the following transition probability matrix P = .4 .2 .4 .6 .3 .1 0 .5 .5 Suppose that the chain starts at either states 2 or 3 with the same probability, in other words, the initial distribution is p0= [0, .5, .5]. Find (i) P(X0= 2) (1 POINT) (ii) P(X1= 2) (2 POINTS) (iii) P(X∞= 2) (2 POINTS) (iii) E3T1 (3 POINTS) (Hint: notice that this matrix, in addition to its rows adding up to 1, also satisfies that its columns add up to 1. This should make finding a left eigenvector for the eigenvalue 1, and therefore, finding the stationary distribution, a very simple task. No MATLAB needed!)
Consider continuous-time Markov chain with a state space {1,2,3} with A1 = 2, A2 = 3, A3 = 4 The underlying discrete transition probabilities are given by P = [0 0.5 0.5; 2/3 0 1/3; 0.5 0.5 0] (a) Find the generator matrix. (b) Find the stationary distribution of this CTMC.
Sri K.
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