# Using properties of Poisson process

I'm trying to make sure I understand the Poisson process.

Let $N(t)$ and $M(t)$ be independent Poisson processes with rates $2,3$ respectively. Find the expected time until the first jump in $N(t)$ after the 2nd jump in $M(t)$.

Is it as simple as using the memoryless property of the exponential (and independence of $N(t), M(t)$) to conclude that the expected wait time is $\frac{1}{2}$?

• Almost...it is 1/2 after that second jump in $M$ happened. (I assume there is a typo.) – Ian May 14 '18 at 3:29
• My apologies. Yes it should be $M(t)$. – Supdawg314 May 14 '18 at 3:38

The grammar in the problem statement is slightly ambiguous, but my interpretation is "find the expected time until the first jump in $N(t)$ which occurs after the second jump in $M(t)$, measuring the time starting from the start of the process". In this case, by the memoryless property and the independence, this is the sum of the expected time of the second jump in $M(t)$ and the expected time of the first jump in $N(t)$, so the total is $2/3+1/2$.