For a Q matrix representing the transitions between transient states in an absorbing Markov chain,...

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For a Q matrix representing the transitions between transient states in an absorbing Markov chain, it can be shown that (I - Q) -1 = I + Q + Q 2 + · · · + Q n + · · · a. Explain why this expression for (I - Q) -1 is plausible. b. Define m ij = expected number of periods spent in transient state t j before absorption, given that we begin in state t i . (Assume that the initial period is spent in state t i .) Explain why m ij + (probability that we are in state t j initially) + (probability that we are in state t j after first transition) + (probability that we are in state t j after second transition) + · · · +(probability that we are in state t j after nth transition) + · · ·. c. Explain why the probability that we are in state t j initially = ijth entry of the (s - m) × (s - m) identity matrix. Explain why the probability that we are in state t j after nth transition = ijth entry of Q n . d. Now explain why m ij = ijth entry of (I - Q) -1 . Aug 29 2014 11:56 AM

 

Solution ID:609225 | This paper was updated on 26-Nov-2015

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