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 $N_2$?