(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)
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...