De Moivre–Laplace theorem

Statement

Let $p \in (0, 1)$ and let $S_n \sim \operatorname{Bin}(n, p)$ for each $n \ge 1$. Let $q = 1 - p$. Then $(S_n - np)/\sqrt{npq}$ converges in distribution to $\mathcal{N}(0, 1)$. That is, for all reals $a < b$, $$\lim_{n \to \infty} P\left(a \le \frac{S_n - np}{\sqrt{npq}} \le b\right) = \int_a^b \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx.$$

Remarks

This is the special case of the central limit theorem for Bernoulli variables. Indeed, $S_n$ has the law of a sum of $n$ independent Bernoulli($p$) variables. A local form also holds: $\binom{n}{k} p^k q^{n-k} \sim \frac{1}{\sqrt{2\pi npq}} e^{-(k - np)^2/(2npq)}$ as $n \to \infty$, uniformly for $k$ with $|k - np| \le C\sqrt{n}$. The proof uses Stirling's formula. In practice, one often adds a continuity correction of $1/2$ to the integer bounds.

History

Abraham de Moivre proved the case $p = 1/2$. Pierre-Simon Laplace extended it to general $p$.

Explanations

L1 human

For many trials, the binomial distribution looks like a bell curve with mean $np$ and variance $np(1 - p)$. This lets you approximate binomial probabilities, such as "at least $520$ heads in $1000$ tosses", with the normal distribution.