# Inter-event time between 1st and 2nd events for Poisson Process

We have two independent Poisson process $N_1$ and $N_2$ with parameter $\tau_1$ and $\tau_2$. Now I want to determine the inter-arrival time between the 1st and 2nd events. Given that processes are memoryless and have stationary and independent increments.

The answer of this question is: the inerarrival time between the 1st and 2nd event from the two process will have exponential distribution with parameter $\tau_1 + \tau_2$. But I am trying to prove it without merging the Poisson process. How can I find the distribution of the inter-arrival time between the 1st and 2nd events using the memoryless property of the arrival times of $N_1$ and $N_2$?

• The first method in my answer to your previous question will do it. – André Nicolas Jul 31 '15 at 20:49
• So if two consecutive arrivals both come from the first process, then that's not what you're looking for? The question would seem to need a bit of clarification. ${}\qquad{}$ – Michael Hardy Jul 31 '15 at 20:51
• Actually, it does not matter from which process its coming. we will count only the arrivals. actually it might happen that both 1st and 2nd come from the 1st process only bcz of the memoryless property after every event it will be in the same condition. I have updated the question already. please check. – jhon_wick Jul 31 '15 at 21:10