## The Poisson Distribution

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:

• The number of cars that pass through a certain point on a road (sufficiently distant from traffic lights) during a given period of time.
• The number of spelling mistakes one makes while typing a single page.
• The number of phone calls at a call center per minute.
• The number of times a web server is accessed per minute.
• The number of roadkill (animals killed) found per unit length of road.
• The number of mutations in a given stretch of DNA after a certain amount of radiation.
• The number of pine trees per unit area of mixed forest.
• The number of stars in a given volume of space.
• The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
• The number of light bulbs that burn out in a certain amount of time.
• The number of viruses that can infect a cell in cell culture.
• The number of inventions invented over a span of time in an inventor's career.
• The number of particles that "scatter" off of a target in a nuclear or high energy physics experiment.