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: 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