Probability Theory
"Poisson theorem" redirects here. For the "Poisson's theorem" in Hamiltonian mechanics, see Poisson bracket § Constants of motion.
In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions.[1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem.
For broader coverage of this topic, see Poisson distribution § Law of rare events.
Theorem[]
Let
be a sequence of real numbers in
such that the sequence
converges to a finite limit
. Then:
![{\displaystyle \lim _{n\to \infty }{n \choose k}p_{n}^{k}(1-p_{n})^{n-k}=e^{-\lambda }{\frac {\lambda ^{k}}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2a3853c310482f84dda7f07f5f99f308e75bcc)
Proofs[]
.
Since
![{\displaystyle \lim _{n\to \infty }\left(1-{\frac {\lambda }{n}}\right)^{n}=e^{-\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84f9e7e4cb9d8e01cabb70e93b41052ae8e4ef93)
and
![{\displaystyle \lim _{n\to \infty }\left(1-{\frac {\lambda }{n}}\right)^{-k}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a46c16acc61dc348da112bff63476055bef37d53)
This leaves
![{\displaystyle {n \choose k}p^{k}(1-p)^{n-k}\simeq {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df85d961c4cf7b4b795a9f3aa72ff850963c7c80)
Alternative proof[]
Using Stirling's approximation, we can write:
![{\displaystyle {\begin{aligned}{n \choose k}p^{k}(1-p)^{n-k}&={\frac {n!}{(n-k)!k!}}p^{k}(1-p)^{n-k}\\&\simeq {\frac {{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}{{\sqrt {2\pi \left(n-k\right)}}\left({\frac {n-k}{e}}\right)^{n-k}k!}}p^{k}(1-p)^{n-k}\\&={\sqrt {\frac {n}{n-k}}}{\frac {n^{n}e^{-k}}{\left(n-k\right)^{n-k}k!}}p^{k}(1-p)^{n-k}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db8039b02df92fe7f77a4d5f0e52e9ec501511f2)
Letting
and
:
![{\displaystyle {\begin{aligned}{n \choose k}p^{k}(1-p)^{n-k}&\simeq {\frac {n^{n}\,p^{k}(1-p)^{n-k}e^{-k}}{\left(n-k\right)^{n-k}k!}}\\&={\frac {n^{n}\left({\frac {\lambda }{n}}\right)^{k}\left(1-{\frac {\lambda }{n}}\right)^{n-k}e^{-k}}{n^{n-k}\left(1-{\frac {k}{n}}\right)^{n-k}k!}}\\&={\frac {\lambda ^{k}\left(1-{\frac {\lambda }{n}}\right)^{n-k}e^{-k}}{\left(1-{\frac {k}{n}}\right)^{n-k}k!}}\\&\simeq {\frac {\lambda ^{k}\left(1-{\frac {\lambda }{n}}\right)^{n}e^{-k}}{\left(1-{\frac {k}{n}}\right)^{n}k!}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad205bebf4f9a1c824eacf8ca77990deca99b145)
As
,
so:
![{\displaystyle {\begin{aligned}{n \choose k}p^{k}(1-p)^{n-k}&\simeq {\frac {\lambda ^{k}e^{-\lambda }e^{-k}}{e^{-k}k!}}\\&={\frac {\lambda ^{k}e^{-\lambda }}{k!}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66670b4d6e131ad53d471ac4653e1285c9d4e517)
Ordinary generating functions[]
It is also possible to demonstrate the theorem through the use of ordinary generating functions of the binomial distribution:
![{\displaystyle G_{\operatorname {bin} }(x;p,N)\equiv \sum _{k=0}^{N}\left[{\binom {N}{k}}p^{k}(1-p)^{N-k}\right]x^{k}={\Big [}1+(x-1)p{\Big ]}^{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84f0051a42e4b4e3ad464aa8519f814360e3697c)
by virtue of the binomial theorem. Taking the limit
while keeping the product
constant, we find
![{\displaystyle \lim _{N\rightarrow \infty }G_{\operatorname {bin} }(x;p,N)=\lim _{N\rightarrow \infty }\left[1+{\frac {\lambda (x-1)}{N}}\right]^{N}=\mathrm {e} ^{\lambda (x-1)}=\sum _{k=0}^{\infty }\left[{\frac {\mathrm {e} ^{-\lambda }\lambda ^{k}}{k!}}\right]x^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20230fc7a78091820f40495f377f27f4e36bb848)
which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the exponential function.)
See also[]
References[]