How to Derive the Distribution of Two Poisson Processes in Series?

[Chinnu]

Homework Statement

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 P_ab(t₁) and the for B to C is t₂ and the distribution is P_bc(t₂). You are given:

P_ab(t₁) = k₁exp(-k₁t₁) and P_bc(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 P_ac(t). Also find the mean of this distribution in terms of k₁ and k₂.

Homework Equations

We can use the basic rules of probability,

The Attempt at a Solution

Is it as simple as this?:

P_ac(t) = k₁P_ab(t₁) + k₂P_bc(t₂)

or is it:

P_ac(t) = k₁k₂exp(-k₁t₁-k₂t₂)

 

[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.