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

Attempt:

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...

 

To find the PDF of Y, you must first find the probability P{Y<t} or P{Y>t}.

I think you are in the right direction you just have to translate the probability of times to the probability of events.

P{Y>t}=P{"No events between times x1 and t"}=P{"Only one event until time t"}=.....


Hope this helps,


Alex