Hello,

If you know no formula for this, the best way is to partition the set of events :

(assuming that N and the sequence X_i are independent, and I guess there's an extra N in what you wrote... anyway, it would be the same reasoning)

And

The sum of Poisson variables is a Poisson variable (whose parameter is the sum of the parameters)

And you have to see what happens if N is 0. This should be stated in your problem, or conventionally, the sum would be 0.

It's a bit messy, but you can get the information you need from here...