The Poisson Distribution

Simeon Denis Poisson
Siméon Denis Poisson

Suppose one expects to find on average $\lambda$ randomly and uniformly distributed objects or independent occurrences of something in a given area, volume, measure of time, etc.

Let $X$ be the random variable that counts how many are actually seen. The probabilities associated with $P(X=x)$ follow what is known as the Poisson Distribution (first introduced by Siméon Denis Poisson in 1837), and has a probability mass function 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 may remember 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: