Another important discrete probability distribution is the **Poisson distribution**, named in
honor of the French mathematician and physicist Simeon D. Poisson (1781-1840). The Poisson
distribution is often used to describe the probability of a number of
events occurring in a given time or space interval, with the probability
of occurrence of these events being very small (Lovric 2011)). However, since the number of trials
is very large, these events do actually occur.

The random variable \(X\), called a
**Poisson random variable**, is considered the number of
occurrences (or arrivals) of such events in a given interval in time or
space. A Poisson random variable has infinitely many possible values,
namely, all whole numbers (integer).

Assuming that \(\lambda\) is the expected value of such arrivals in a time interval of fixed length, the probability of observing exactly \(x\) events is given by the probability mass function

\[ P(X = x) = e^{-\lambda}\frac{\lambda x}{x!}, \qquad x = 0, 1, 2, . . . ,\]

where \(\lambda\) is a positive real number, which represents the average number of events occurring during a fixed time interval, and \(e \approx 2.7182818\).

Thus, any particular Poisson distribution is identified by one parameter, usually denoted \(\lambda\) (the Greek letter lambda). For example, if the event occurs 10 times per second on average, in 60 seconds the event will occur on average 600 times and \(\lambda = 600\).

**Citation**

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: *Hartmann,
K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis
using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.*