Thus transition probability matrix 6 4 chapman

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Unformatted text preview: probability matrix: 6 4. Chapman-Kolmogorov equation So far we have mentioned one-step transition probabilities, i.e., probability of An given equation provides a relation for multiple steps as follows: p n + m = ij where n m=0 12 n C-K np m: kj 1 sp t: 2 ik  k :::. For continuous time, this can be written as: p s + t = ij where s Xp A ,1 . Xp ik  kj k t  0: In matrix notation, P s + t = P sP t: Set the initial transition probability matrix as P 0 = I i.e., p ij 0 Define, q ij =1 (identity matrix) if i = j else 0 if i 6= j: lim p t , p 0 = lim p tt for j 6= i: !0 !0 t  q := lim p t , p 0 = lim p tt , 1 : !0 !0 t := ij ij ij t t ii ii ii ii t t Note that q ’s are instantaneous rate. In matrix notation, Q = lim P tt , I : !0 t where, Q = q : Now, recall the relation ij Xp t = 1: ij  j If we move 1 from RHS to LHS of this equation and then divide by t and let t ! 0, we get lim fp 1t=t + ::::: + p t , 1=t + :::: + p t=t + :::g = 0 !0 t i which implies ii q...
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This document was uploaded on 03/19/2014 for the course CS 6030 at Western Michigan.

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