hw4soln - ORIE 361/523 Homework 4 Instructor Bikramjit Das...

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ORIE 361/523 – Homework 4 Instructor: Bikramjit Das due July 27, 2007 Answers. 1. If Σ i P i,j = 1 j, then, r j = 1 M +1 satisfies r j = Σ m i =0 r i P i,j 1 = Σ m i =0 r i Hence by uniqueness of stationary distribution(since the chain is irreducible and aperiodic), these are the limiting probabilities. 2. Consider the Markov chain which has 2 states, S if the dividend has been suspended and N if it is not. Let, X n = The status of dividend payment on the nth observed period. Then, the transition matrix P is P = S N ± 0 . 6 0 . 4 0 . 1 0 . 9 ² This is clearly an irreducible Markov chain, which has finite state space. So, we know, the long run proportions of staying in state j is π j , where π is the stationary distribution for the Markov chain. We proceed to compute the stationary distribution. πP = π X i π i = 1 gives 0 . 6 π S + 0 . 1 π N = π S 0 . 4 π S + 0 . 9 π N = π N π S + π N = 1 Solving, we get π N = 4 5 , which is the required answer.
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hw4soln - ORIE 361/523 Homework 4 Instructor Bikramjit Das...

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