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First Passage Decomposition

  1. Nov 15, 2011 #1
    1. The problem statement, all variables and given/known data

    I have a Stochastic Processes test coming up soon and we are taught about the first passage decomposition. I understand this but it then says that it changes it to a generating function

    2. Relevant equations
    Generating function Pij(s) = Ʃ pij(n)sn
    and Fij(s) = Ʃ fij(n)sn

    What does this generating function actually mean and what is the difference between
    Pij(s) and Fij(s)???
  2. jcsd
  3. Nov 15, 2011 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Well, pij(n) and fij(n) are different things, so of course their generating functions are different. The generating function does not have to mean anything; it is just a useful tool in the arsenal of problem-solving resources. However in the case of Fij(s) we can assign a meaning, for what it is worth: Fij(s) = E[sTij], where Tij = the first passage time from state i to state j, at least in the case where such passage is certain (i.e., Pr{Tij < ∞} = 1.)

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