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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).
Theorem
Let
be a sequence of real numbers in
such that the sequence
converges to a finite limit
. Then:
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
.
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/4b1a210533bffb9a0cb4cf1b4eb62c2049129be3)
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}(1-{\frac {\lambda }{n}})^{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/7510002a5903bafbf53a88c683e1083245e93fc0)
As
,
so:
Ordinary Generating Functions
It is also possible to demonstrate the theorem through the use of Ordinary Generating Functions of the binomial distribution:
![{\displaystyle G_{\mathrm {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/5f3ce093093f971cffcd41e083e9bf050192af1c)
by virtue of the Binomial Theorem. Taking the limit
while keeping the product
constant, we find
![{\displaystyle \lim _{N\rightarrow \infty }G_{\mathrm {bin} }(x;p,N)=\lim _{N\rightarrow \infty }{\Big [}1+{\frac {\lambda (x-1)}{N}}{\Big ]}^{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/2f0728e1ce9c1b18409921ed1e21c8690c17eac0)
which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the Exponential function.)
See also
References
- ^ Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, 4th Edition