Probability theory and stochastic processes Limit theorems Theorem

De Moivre–Laplace theorem

Builds on Binomial distribution Normal distribution Convergence in distribution Central limit 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 $p$ .

Reading path

To understand this page from the ground up, read these 71 pages in order. The foundations come first.

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The rationals form an ordered field statement
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Density of the normal distribution definition
Characteristic function definition
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Lévy's continuity theorem statement
Central limit theorem statement
De Moivre–Laplace theorem statement

Tag	`065X9S8CM3`
URI	`m2://065X9S8CM3`
Aliases	`de-moivre-laplace-theorem`, `normal-approximation-to-the-binomial`
Kind	statement, theorem
Writing	W1 stated
MSC	60F05
Status	This copy of the site has no build data. Run `m2 status 065X9S8CM3`.
Uses	Binomial distribution, Normal distribution, Convergence in distribution, Central limit theorem
Same as	Wikidata Q1855610 Wikipedia De Moivre–Laplace theorem

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