Describe the Poisson distribution characterised by parameter $\lambda>0$. Calculate the mean and variance of this distribution in terms of $\lambda$.

Show that the sum of $n$ independent random variables, each having the Poisson distribution with $\lambda=1$, has a Poisson distribution with $\lambda=n$.

Use the central limit theorem to prove that

$$e^{-n}\left(1+\frac{n}{1 !}+\frac{n^{2}}{2 !}+\ldots+\frac{n^{n}}{n !}\right) \rightarrow 1 / 2 \quad \text { as } \quad n \rightarrow \infty$$