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 2010). 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 (Weiss 2010).

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 on average 10 times per second, in 60 seconds the event will occur on average 600 times and \(\lambda = 600\).