Suppose that each month 10 of transit users go back

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Unformatted text preview: /3 0 2/3 0 0 1/6 0 5/6 2 /5 0 3/5 0 and show that it does not satisfy the detailed balance condition (1.11). (b) Consider 1 2 3 4 1 0 a 0 1a 21b 0 b 0 3 0 1c 0 c 4 d 0 1d 0 and show that there is a stationary distribution satisfying (1.11) if 0 < abcd = (1 a)(1 b)(1 c)(1 d). 1.13. Consider the Markov chain with transition matrix: 1 2 3 4 1 0 0 0.8 0.4 2 0 0 0.2 0.6 3 0.1 0.6 0 0 4 0.9 0.4 0 0 (a) Compute p2 . (b) Find the stationary distributions of p and all of the stationary distributions of p2 . (c) Find the limit of p2n (x, x) as n ! 1. 1.14. Do the following Markov chains converge to equilibrium? (a) 1 1 0 2 0 3 .3 4 1 2 0 0 .7 0 3 1 .5 0 0 4 0 .5 0 0 (b) 1 1 0 2 0 3 1 4 1/3 23 1 .0 0 0 0 0 0 2/3 4 0 1 0 0 65 1.12. EXERCISES (c) 1 10 20 30 41 50 6 .2 2 .5 0 0 0 1 0 3 .5 0 0 0 0 0 4 0 1 .4 0 0 0 5 0 0 0 0 0 .8 6 0 0 .6 0 0 0 1.15. Find limn!1 pn (i, j ) for 1 2 p= 3 4 5 1 2 3 4 5 1 0 0 0 0 0 2/3 0 1 /3 0 1/8 1/4 5/8 0 0 0 1 /6 0 5 /6 0 1/3 0 1/3 0 1/3 You are supposed to do this and the next problem by solving...
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