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Poisson distribution

In statistics and probability theory, the Poisson distribution is a discrete probability distribution (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) belonging to 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. The probability that there are exactly k occurrences (k being a natural number including 0, k = 0, 1, 2, ...) is:

Where: Sometimes λ is taken to be the rate, i.e., the average number of occurrences per unit time. In that case, if Nt is the number of occurrences before time t then we have
and the waiting time T until the first occurrence is a continuous random variable with an exponential distribution; this probability distribution may be deduced from the fact that

Table of contents
1 Occurrence
2 How does this distribution arise? -- The limit theorem
3 Properties


The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete nature (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples include:

How does this distribution arise? -- The limit theorem

The binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity.


The expected value of a Poisson distributed random variable is equal to λ and so is its variance.

The most likely value ("mode") of a Poisson distributed random variable is equal to the largest integer ≤ λ, which is also written as floor(λ).

If λ is big enough (λ > 10 say), then the normal distribution with mean λ and standard deviation √ λ is an excellent approximation to the Poisson distribution.

If N and M are two independent random variables, both following a Poisson distribution with parameters λ and μ, respectively, then N + M follows a Poisson distribution with parameter λ + μ.

The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the law of small numbers because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is the title of a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898. Some historians of mathematics have argued that the Poisson distribution should have been called the Bortkiewicz distribution.