# Poisson distribution

2010 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A probability distribution of a random variable taking non-negative integer values with probabilities

where is a parameter. The generating function and the characteristic function of the Poisson distribution are defined by

respectively. The mean, variance and the semi-invariants of higher order are all equal to . The distribution function of the Poisson distribution,

is given at the points by

where is the value at the point of the gamma-distribution function with parameter (or by , where is the value at the point of the "chi-squared" distribution function with degrees of freedom) whence, in particular,

The sum of independent variables each having a Poisson distribution with parameters has a Poisson distribution with parameter .

Conversely, if the sum of two independent random variables and has a Poisson distribution, then each random variable and is subject to a Poisson distribution (Raikov's theorem). There are general necessary and sufficient conditions for the convergence of the distribution of sums of independent random variables to a Poisson distribution. In the limit, as , the random variable has the standard normal distribution.

The Poisson distribution was first obtained by S. Poisson (1837) when deriving approximate formulas for the binomial distribution when (the number of trials) is large and (the probability of success) is small. See Poisson theorem 2). The Poisson distribution describes many physical phenomena with good approximation (see [F], Vol. 1, Chapt. 6). The Poisson distribution is the limiting case for many discrete distributions such as, for example, the hypergeometric distribution, the negative binomial distribution, the Pólya distribution, and for the distributions arising in problems about the arrangements of particles in cells with a given variation in the parameters. The Poisson distribution also plays an important role in probabilistic models as an exact probability distribution. The nature of the Poisson distribution as an exact probability distribution is discussed more fully in the theory of random processes (see Poisson process), where the Poisson distribution appears as the distribution of the number of certain random events occurring in the course of time in a fixed interval:

(the parameter is the mean number of events in unit time), or, more generally, as the distribution of a random number of points in a certain fixed domain of Euclidean space (the parameter of the distribution is proportional to the volume of the domain).

Along with the Poisson distribution, as defined above, one considers the so-called generalized or compound Poisson distribution. This is the probability distribution of the sum of a random number of identically-distributed random variables (where are considered to be mutually independent and is distributed according to the Poisson distribution with parameter ). The characteristic function of the compound Poisson distribution is

where is the characteristic function of . For example, the negative binomial distribution with parameters and is a compound Poisson distribution, since one can put

The compound Poisson distributions are infinitely divisible and every infinitely-divisible distribution is a limit of compound Poisson distributions (perhaps "shifted" , that is, with characteristic functions of the form ). In addition, the infinitely-divisible distributions (and these alone) can be obtained as limits of the distributions of sums of the form , where form a triangular array of independent random variables each with a Poisson distribution, and where and are real numbers.

#### References

 [P] S.D. Poisson, "Récherches sur la probabilité des jugements en matière criminelle et en matière civile", Paris (1837) [F] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1950–1966) [BS] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics", Libr. math. tables, 46, Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) Zbl 0529.62099 [LO] Yu.V. Linnik, I.V. Ostrovskii, "Decomposition of random variables and vectors", Amer. Math. Soc. (1977) (Translated from Russian) MR0428382 Zbl 0358.60020