The Poisson Distribution

When one expects to find on average some number of randomly and uniformly distributed objects or occurrences (that are independent of one another) of something in a given area, volume, measure of time, etc..., and one wants to find the probability of seeing exactly $X$ objects/occurrences (which may be different than the expected number), the Poisson Distribution is used.

If the expected number of objects/occurrences described above is denoted by the Greek letter, lambda, then the probability of seeing exactly $X$ objects/occurrences is given by

$$P(X) = \frac{e^{-\lambda} \lambda^X}{X!}$$

Recall $e$ is an important constant in mathematics. If you are familiar with calculus you will recall that $$e = \lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$ For the purpose of calculating Poisson probabilities, however, it is sufficient to know that its value is approximately given by $$e \approx 2.718$$

There are many examples of when using the Poisson distribution might be appropriate: