Define what it means for a random variable $X$ to have a Poisson distribution, and find its moment generating function.

Suppose $X, Y$ are independent Poisson random variables with parameters $\lambda, \mu$. Find the distribution of $X+Y$.

If $X_{1}, \ldots, X_{n}$ are independent Poisson random variables with parameter $\lambda=1$, find the distribution of $\sum_{i=1}^{n} X_{i}$. Hence or otherwise, find the limit of the real sequence

$$a_{n}=e^{-n} \sum_{j=0}^{n} \frac{n^{j}}{j !}, \quad n \in \mathbb{N}$$

[Standard results may be used without proof provided they are clearly stated.]