# law of rare events

Let $X$ be distributed as $Bin(n,p)$, a binomial random variable with parameters $n$ and $p$. Suppose

 $\lim_{n\rightarrow\infty}np=\lambda,$

where $\lambda$ is a positive real constant, then $X$ is asymptotically distributed as $Poisson(\lambda)$, a Poisson distribution with parameter $\lambda$.

Basically, when the size of the population $n$ is very large and the occurrence of certain event $A$ is rare, where $p$, the probability of $A$ is very small, the binomial random variable $X$ can be approximated by a Poisson random variable.

Sketch of Proof. Let $X\sim Bin(n,p)$. So

 $\displaystyle P(X=m)$ $\displaystyle=$ $\displaystyle\frac{n!}{m!(n-m)!}p^{m}(1-p)^{n-m}$ $\displaystyle=$ $\displaystyle\frac{n!}{n^{m}(n-m)!}\frac{(np)^{m}}{m!}(1-\frac{np}{n})^{n-m}$ $\displaystyle=$ $\displaystyle\frac{n!}{n^{m}(n-m)!}\frac{(np)^{m}}{m!}(1-\frac{np}{n})^{n}(1-% \frac{np}{n})^{-m}.$

As $n\rightarrow\infty$,

 $\frac{n!}{n^{m}(n-m)!}=\frac{n}{n}\frac{n-1}{n}\cdots\frac{n-m+1}{n}\approx 1,$
 $(1-\frac{np}{n})^{-m}\approx(1-\frac{\lambda}{n})^{-m}\approx 1,$
 $(1-\frac{np}{n})^{n}\approx(1-\frac{\lambda}{n})^{n}\approx e^{-\lambda},$

and

 $\frac{(np)^{m}}{m!}\approx\frac{\lambda^{m}}{m!}.$

Therefore,

 $P(X=m)\approx\frac{\lambda^{m}}{m!}e^{-\lambda}=Poisson(\lambda).$

Example. Suppose in a given year, the number of fatal automobile accidents has a binomial distribution for a particular insuarance company with five hundred automobile insurance policies. On average, there is one policy out of the five hundred that will be involved in a fatal crash. What is the probability that there will be no fatal accidents (out of five hundred policies) in any particular year?

Solution. If $X$ be the number of fatal accidents in a year from a population of 500 auto policies, then $X\sim Bin(n,p)$ with $n=500$ and $p=1/500$. $\lambda=500\times 1/500=1$ and so

 $P(X=0)\approx e^{-1}\approx 0.368.$

Using the binomial distribution, we have

 $P(X=0)=(1-\frac{1}{500})^{500}\approx 0.367.$
Title law of rare events LawOfRareEvents 2013-03-22 14:39:32 2013-03-22 14:39:32 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 62P05 msc 60E99 msc 60F99 Poisson theorem