# Two Poisson processes in Series

1. Sep 25, 2011

### Chinnu

1. The problem statement, all variables and given/known data

Imagine you want to go from A to C via B. So you have two steps: A to B and B to C. Let's assume the time taken (t1) to go from A to B is Poisson and is given by Pab(t1) and the for B to C is t2 and the distribution is Pbc(t2). You are given:

Pab(t1) = k1exp(-k1t1) and Pbc(t2) = k2exp(-k2t1)

So the total time to go from A to C is t = t1 + t2

Derive the distribution of t, that is, find Pac(t). Also find the mean of this distribution in terms of k1 and k2.

2. Relevant equations

We can use the basic rules of probability,

3. The attempt at a solution

Is it as simple as this?:

Pac(t) = k1Pab(t1) + k2Pbc(t2)

or is it:

Pac(t) = k1k2exp(-k1t1-k2t2)

Last edited: Sep 25, 2011
2. Sep 26, 2011

### Chinnu

I am leaning towards the second equation I wrote above, as, if you take the case that k1 = k2 = 1, then the resulting equation is that of a normal poisson with t = t1 + t2.

I only want to know if this thinking is correct, I know how to continue from there.