Poisson limit theorem

Statement

Let $\lambda > 0$, and let $p_n \in [0, 1]$ satisfy $n p_n \to \lambda$ as $n \to \infty$. Let $S_n \sim \operatorname{Bin}(n, p_n)$. Then for each integer $k \ge 0$, $$\lim_{n \to \infty} P(S_n = k) = \lim_{n \to \infty} \binom{n}{k} p_n^k (1 - p_n)^{n - k} = e^{-\lambda} \frac{\lambda^k}{k!}.$$ So $S_n$ converges in distribution to the Poisson distribution with parameter $\lambda$.

Remarks

Proof: write $\binom{n}{k} p_n^k = \frac{n(n-1)\cdots(n-k+1)}{n^k} \cdot \frac{(n p_n)^k}{k!}$. The first factor tends to $1$, and $(1 - p_n)^{n-k} \to e^{-\lambda}$. In practice, $\operatorname{Bin}(n, p)$ is close to $\operatorname{Poisson}(np)$ when $n$ is large and $p$ is small. Le Cam's inequality gives an explicit bound on the error of this approximation.

Explanations

L1 human

The number of rare events among many independent chances is approximately Poisson. Examples are the number of misprints on a page, or the number of large insurance claims in a year.