The Poisson Distribution

The Binomial distribution needs a precisely known, finite number of trials $n$, together with the probability of success on each one. But many real counting problems don't fit that mould at all: the number of phone calls arriving at an exchange in an hour, the number of printing errors on a page, the number of road accidents in a city in a day, the number of snakebite cases reported in a locality in a year. In situations like these we know how many times the event did happen, but there is no natural, fixed "number of trials" and no way to count how many times it did not happen — so $n$ and $p$ individually are not really meaningful, even though the event is genuinely rare on a per-opportunity basis. The Poisson distribution is built exactly for this kind of rare, randomly-occurring count.

Definition. A discrete random variable $X$ taking values $0, 1, 2, 3, \dots$ is said to follow a Poisson distribution with parameter $\lambda > 0$ if

$P(X = x) = \dfrac{e^{-\lambda}\lambda^x}{x!}, \qquad x = 0, 1, 2, \dots$

Here $\lambda$ is the single parameter of the distribution — it represents the average (expected) number of occurrences of the event over the fixed interval of time, space, or opportunity being considered. Poisson probabilities are used when: each trial has two possible outcomes (occurrence / non-occurrence); the number of underlying "trials" is very large; the trials are independent; and the probability of occurrence on any one of them is very small — exactly the regime where a Binomial count becomes impossible to track individually but still has a well-defined average rate.

Poisson as a limit of the Binomial. This connection can be made precise: if $n$ is allowed to grow indefinitely while $p = \lambda/n$ shrinks in exact proportion (so that the mean $np = \lambda$ stays fixed), then the Binomial probability $P(X=x) = {}^{n}C_x\,p^x q^{\,n-x}$ converges, term by term, to the Poisson probability $\dfrac{e^{-\lambda}\lambda^x}{x!}$ as $n \to \infty$. In other words, the Poisson distribution is what a Binomial distribution turns into when there are effectively unlimited opportunities for a very rare event to occur, but the average count over the interval stays at a fixed, finite value $\lambda$. This is why the Poisson distribution is the natural stand-in for the Binomial exactly in the "large $n$, tiny $p$" situations described above.

Mean and variance. A special and very convenient feature of the Poisson distribution is that its mean and its variance are equal, and both equal the parameter itself: $\mu = \sigma^2 = \lambda$. So the single number $\lambda$ simultaneously tells us the expected count and how much that count typically varies. …