Basic Poisson expectation question

Question

I'm trying to grasp poisson processes once again after taking a statistics course a long time ago.

I want to find $E(X|X>2)$ where $X \sim poi(3)$

My solution so far:

$E(X)=\lambda = rt$ where $r=$ the average rate at which the events occur and $t=$ time interval.

This would have solve the problem if we were given a specific time interval, but the exercise gives just gives me $X>2$ with no upper limit, so therefore I don't know where to go from this.

Would anyone like to give me hint?

Answer 1

$E(X|X>2)= \frac {\sum\limits_{n=3}^{\infty} ne^{-3} \frac {3^{n}} {n!}} {\sum\limits_{n=3}^{\infty} e^{-3} \frac {3^{n}} {n!}}$.