---
tag: 065X9S8CM3
kind: statement
role: theorem
name: De Moivre–Laplace theorem
aliases: [de-moivre-laplace-theorem, normal-approximation-to-the-binomial]
msc: [60F05]
uses: [0VRWZDWW6B, 07RXBVTVZB, 0JDAQHWA9N, 0EPZC4Z1FC]
sources:
  - url: https://en.wikipedia.org/wiki/De_Moivre–Laplace_theorem
    locator: Theorem
same_as:
  wikidata: Q1855610
  wikipedia: 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|0JDAQHWA9N]] 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|0EPZC4Z1FC]] 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.