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.

Tag	`0XKTZRWTXX`
URI	`m2://0XKTZRWTXX`
Aliases	`law-of-rare-events`, `poisson-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 0XKTZRWTXX`.
Uses	Binomial distribution, Poisson distribution, Convergence in distribution
Same as	Wikidata Q7208500 Wikipedia Poisson limit theorem

Tag

0XKTZRWTXX

URI

m2://0XKTZRWTXX

Aliases

law-of-rare-events, poisson-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 0XKTZRWTXX.

Uses

Binomial distribution, Poisson distribution, Convergence in distribution

Same as

Wikidata Q7208500

Wikipedia Poisson limit theorem

Poisson limit theorem

Statement

Remarks

Reading path

Further reading

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