Poisson distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A probability distribution of a random variable $ X $ taking non-negative integer values $ k = 0 , 1 \dots $ with probabilities

$$ {\mathsf P} \{ X = k \} = e ^ {- \lambda } \frac{\lambda ^ {k} }{k!} , $$

where $ \lambda > 0 $ is a parameter. The generating function and the characteristic function of the Poisson distribution are defined by

$$ \phi ( z) = e ^ {\lambda ( z - 1 ) } \ \ \textrm{ and } f ( t) = \ \mathop{\rm exp} [ \lambda ( e ^ {it} - 1 ) ] , $$

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

$$ F ( x) = \sum _ { i= } 0 ^ { [ } x] e ^ {- \lambda } \frac{\lambda ^ {i} }{i!} , $$

is given at the points $ k = 0 , 1 ,\dots $ by

$$ F ( k) = \frac{1}{k!} \int\limits _ \lambda ^ \infty y ^ {k} e ^ {- \lambda } d y = 1 - S _ {k+} 1 ( \lambda ) , $$

where $ S _ {k+} 1 ( \lambda ) $ is the value at the point $ \lambda $ of the gamma-distribution function with parameter $ k + 1 $( or by $ F ( k) = 1 - H _ {2k+} 2 ( 2 \lambda ) $, where $ H _ {2k+} 2 ( 2 \lambda ) $ is the value at the point $ 2 \lambda $ of the "chi-squared" distribution function with $ 2 k + 2 $ degrees of freedom) whence, in particular,

$$ {\mathsf P} \{ X = k \} = S _ {k} ( \lambda ) - S _ {k-} 1 ( \lambda ) . $$

The sum of independent variables $ X _ {1} \dots X _ {n} $ each having a Poisson distribution with parameters $ \lambda _ {1} \dots \lambda _ {n} $ has a Poisson distribution with parameter $ \lambda _ {1} + \dots + \lambda _ {n} $.

Conversely, if the sum $ X _ {1} + X _ {2} $ of two independent random variables $ X _ {1} $ and $ X _ {2} $ has a Poisson distribution, then each random variable $ X _ {1} $ and $ X _ {2} $ 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 $ \lambda \rightarrow \infty $, the random variable $ ( X - \lambda ) / \sqrt \lambda $ has the standard normal distribution.

The Poisson distribution was first obtained by S. Poisson (1837) when deriving approximate formulas for the binomial distribution when $ n $( the number of trials) is large and $ p $( 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 $ X ( t) $ of certain random events occurring in the course of time $ t $ in a fixed interval:

$$ P \{ X ( t) = k \} = e ^ {- \lambda t } \frac{( \lambda t ) ^ {k} }{k!} ,\ \ k = 0 , 1 ,\dots $$

(the parameter $ \lambda $ 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 $ X _ {1} + \dots + X _ \nu $ of a random number $ \nu $ of identically-distributed random variables $ X _ {1} , X _ {2} ,\dots $( where $ \nu , X _ {1} , X _ {2} \dots $ are considered to be mutually independent and $ \nu $ is distributed according to the Poisson distribution with parameter $ \lambda $). The characteristic function $ \phi ( t) $ of the compound Poisson distribution is

$$ \phi ( t) = \mathop{\rm exp} \{ \lambda ( \psi ( t) - 1 ) \} , $$

where $ \psi ( t) $ is the characteristic function of $ X _ \nu $. For example, the negative binomial distribution with parameters $ n $ and $ p $ is a compound Poisson distribution, since one can put

$$ \psi ( t) = \frac{1} \lambda \ \mathop{\rm log} \frac{1}{1 - q e ^ {it} } ,\ \ \lambda = \mathop{\rm log} \ \frac{1}{p} ,\ q = 1 - p . $$

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 $ \mathop{\rm exp} ( \lambda _ {n} ( \psi _ {n} ( t) - 1 - i t a _ {n} )) $). In addition, the infinitely-divisible distributions (and these alone) can be obtained as limits of the distributions of sums of the form $ h _ {n1} X _ {n1} + \dots + h _ {nk _ {n} } X _ {nk _ {n} } - A _ {n} $, where $ ( X _ {n1} \dots X _ {nk _ {n} } ) $ form a triangular array of independent random variables each with a Poisson distribution, and where $ h _ {nk _ {n} } > 0 $ and $ A _ {n} $ 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

Comments

The Poisson distribution frequently occurs in queueing theory.

References

[JK]	N.L. Johnson, S. Kotz, "Distributions in statistics: discrete distributions" , Houghton Mifflin (1970) MR0268996 Zbl 0292.62009