Thursday, July 22, 2010

The Poisson distribution as a limiting form of the Binomial Distribution

This entry is a mirror of the article in Digital Explorations. As you can see, the Wordpress Latex plugin works better compared to the Blogger plugin.

Simeon Dennis Poisson derived his distribution as a limiting form of the binomial distribution
as $$n \to \infty$$ while $$\lambda = n p $$ stays constant. It is very instructive for aspiring
mathematical statisticians to see the details.

We have $$p = \lambda/ n$$ and $$q = 1 - \lambda/n$$. The binomial distribution pmf
$$f(x)=\binom{n}{x} p^x q^{n-x}$$
then becomes $$\frac{n!}{x!(n-x)!} (\frac{\lambda}{n})^x (1 - \frac{\lambda}{n})^{n-x}$$ which when simplifying becomes $$\underbrace{\frac{n!}{x!(n-x)!}}_A \underbrace{(\frac{\lambda}{n})^x}_B
\underbrace{(1 - \frac{\lambda}{n})^{n-x}}_C$$.

Recall that the limit of a product is the
product of the limits of its factors. Consider C:
$$ C = (1 - \frac{\lambda}{n})^{n}(1 - \frac{\lambda}{n})^{-x}$$
As $$n\to \infty, (1 - \frac{\lambda}{n})^{n}$$ converges to $$e^{-\lambda}$$.
while $$(1 - \frac{\lambda/n})^{-x}$$ converges to 1.

Thus we have at the moment,
$$ A B e^{-\lambda}$$
Now consider $$A = \frac{n!}{x!(n-x)!}$$. This can be rewritten as
$$\frac{n(n-1) \cdots (n-k+1)(n-k)(n-k-1)\cdots 2 \cdot 1}{x! (n-x)!}$$ or
$$A= \frac{n(n-1) \cdots (n-(x+1))}{x!}$$ the numerator has x factors.
and factoring,
$$ A = \frac{n^k (1 (1 - 1/n) (1 - 2/n)\cdots (1 - (n-x+1)/n}{x!}
The limit as $$n\to\infty$$ is thus $$A = \frac{n^x(1)1)\cdots(1)}{x!}$$
Putting together ABC, we now have

$$[\frac{n^x}{x!}][\frac{\lambda^n} {n^x}][e^{-\lambda}]$$

From which we have the pmf of the Poisson distribution
$$f(x) = \frac{ \lambda^x e^{-\lambda}}{x!}$$

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