In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. The Poisson distribution can also be used for other specified intervals such as: distance, area or volume. A classic example is the nuclear decay of atoms. The Poisson distribution can be applied to systems with a large number of discrete states and a small number of favourable outcomes. If n is much less than N, then the probability p=n/N approaches 0. Such is the case in the nuclear decay of atoms. In such a case the Poisson distribution can be used to analyse the following problem: Assume that in 1 minute, a sample exhibits 100 radioactive decays. Calculate the probability of finding 0, 1, 2, 3 and 4 decays in any 1-second interval.

The distribution was discovered by Siméon-Denis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile (“Research on the Probability of Judgments in Criminal and Civil Matters”). The work focused on certain random variables N that count, among other things, a number of discrete occurrences (sometimes called “arrivals”) that take place during a time-interval of given length. If the expected number of occurrences in this interval is lambda, then the probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, …) is equal to

f(x,lambda) = (lambda^xe^(-lambda))/x!

where

• e is the base of the natural logarithm (e = 2.71828…)
• x is the number of occurrences of an event - the probability of which is given by the function
• x! is the factorial of x
• lambda is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with lambda = 10/4 = 2.5.

As a function of x, this is the probability mass function. The Poisson distribution can be derived as a limiting case of the binomial distribution. The Poisson distribution is sometimes called a Poissonian, analogous to the term Gaussian for a Gauss or normal distribution.

support x = 0,1,2,3,…