If $X_i, i = 1,...,n$, is Poisson with mean $1$, why is
$$\mathbb P\left\{\sum_{i=1}^n X_i \leq n\right\} = e^{-n}\sum_{k = 0}^n \frac{n^k}{k!}$$
I would appreciate some help/hint here... I recognise the right side of the equation as the CDF of Poisson. The left side, well, it is the probability that the sum is less than (or equal to) n, but why do that equals the CDF?