Negative binomial distribution
A probability distribution of a random variable which takes non-negative integer values
in accordance with the formula
![]() | (*) |
for any real values of the parameters and
. The generating function and the characteristic function of a negative binomial distribution are defined by the formulas
![]() |
and
![]() |
respectively, where . The mathematical expectation and variance are equal, respectively, to
and
. The distribution function of a negative binomial distribution for the values
is defined in terms of the values of the beta-distribution function at a point
by the following relation:
![]() |
where is the beta-function.
The origin of the term "negative binomial distribution" is explained by the fact that this distribution is generated by a binomial with a negative exponent, i.e. the probabilities (*) are the coefficients of the expansion of in powers of
.
Negative binomial distributions are encountered in many applications of probability theory. For an integer , the negative binomial distribution is interpreted as the distribution of the number of failures before the
-th "success" in a scheme of Bernoulli trials with probability of "success"
; in this context it is usually called a Pascal distribution and is a discrete analogue of the gamma-distribution. When
, the negative binomial distribution coincides with the geometric distribution. The negative binomial distribution often appears in problems related to the randomization of the parameters of a distribution; for example, if
is a random variable having, conditionally on
, a Poisson distribution with random parameter
, which in turn has a gamma-distribution with density
![]() |
then the marginal distribution of will be a negative binomial distribution with parameters
and
. The negative binomial distribution serves as a limiting form of a Pólya distribution.
The sum of independent random variables which have negative binomial distributions with parameters
and
, respectively, has a negative binomial distribution with parameters
and
. For large
and small
, where
, the negative binomial distribution is approximated by the Poisson distribution with parameter
. Many properties of a negative binomial distribution are determined by the fact that it is a generalized Poisson distribution.
References
[1] | W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1950–1966) |
Comments
See also Binomial distribution.
References
[a1] | N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions" , Wiley (1969) |
Negative binomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_binomial_distribution&oldid=47951