Poisson limit theorem

The Poisson theorem gives a Poisson approximation to the binomial distribution, under certain conditions. [Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes", 4th Edition]

The theorem

If

:$n\; ightarrow\; infty,\; p\; ightarrow\; 0$, such that $np\; ightarrow\; lambda$

then

:$frac\{n!\}\{(n-k)!k!\}\; p^k\; (1-p)^\{n-k\}\; ightarrow\; e^\{-lambda\}frac\{lambda^k\}\{k!\}.$

Example

Suppose that in an interval of length 1000, 500 points are placed randomly. Now what is the number points that will be placed in a sub-interval of length 10?. If we look here, the probability that a random point will be placed in the sub-interval is $p\; =\; 10/1000\; =\; 0.01$. Here $n=500$ so that $np=5$. The probabilistically precise way of describing the number of points in the sub-interval would be to describe it as a binomial distribution $p\_n(k)$. That is, the probability that $k$ points lie in the sub-interval is

:$p\_n(k)=frac\{n!\}\{(n-k)!k!\}\; p^k\; (1-p)^\{n-k\}.$

But using the Poisson Theorem we can approximate it as

:$e^\{-lambda\}frac\{lambda^k\}\{k!\}\; =\; e^\{-5\}frac\{5^k\}\{k!\}.$

See also

* De Moivre–Laplace theorem

References