Solving Poisson Distribution Homework: Find f(y)

[superwolf]

Homework Statement

In a Poisson process with intensity λ, let X1 be the time until the first event and let X2 be the time between the first and the second event. Let Y be the time until the second event, that is, Y = X1 + X2. Find the density function f(y).

2. The attempt at a solution

Probability that no events occur in time y:

$$p(0; \lambda X1) = e^{- \lambda t}$$

I don't know if this will be helpful at all...

 

[Focus]

I'll give you a hint, the holding times (time between two jumps) of a Poisson process are independent and exponential with parameter lambda. You may want to prove this.