# First Passage Decomposition

## Homework Statement

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

## Homework 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)???

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Ray Vickson
Homework Helper
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## Homework Statement

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

## Homework 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)???
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.)

RGV