# First Passage Decomposition

• stukbv
In summary, the conversation discusses the use of generating functions in the context of Stochastic Processes and the first passage decomposition. The equations for two different generating functions, Pij(s) and Fij(s), are provided and the difference between them is questioned. The expert summarizer explains that the generating function is simply a useful tool and in the case of Fij(s), it can be assigned a specific meaning.

## 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)?

stukbv said:

## 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