Negative binomial distribution

A probability distribution of a random variable $ X $ which takes non-negative integer values $ k = 0, 1 \dots $ in accordance with the formula

$$ \tag{* } {\mathsf P} \{ X = k \} = \ \left ( \begin{array}{c} r+ k- 1 \\ k \end{array} \right ) p ^ {r} ( 1- p) ^ {k} $$

for any real values of the parameters $ 0 < p < 1 $ and $ r > 0 $. The generating function and the characteristic function of a negative binomial distribution are defined by the formulas

$$ P( z) = p ^ {r} ( 1- qz) ^ {-} r $$

and

$$ f( t) = p ^ {r} ( 1- qe ^ {it} ) ^ {-} r , $$

respectively, where $ q = 1- p $. The mathematical expectation and variance are equal, respectively, to $ rq= p $ and $ rq/p ^ {2} $. The distribution function of a negative binomial distribution for the values $ k = 0, 1 \dots $ is defined in terms of the values of the beta-distribution function at a point $ p $ by the following relation:

$$ F( k) = {\mathsf P} \{ X < k \} = \ \frac{1}{B( r, k+ 1) } \int\limits _ { 0 } ^ { p } x ^ {r-} 1 ( 1- x) ^ {k} dx, $$

where $ B( r, k+ 1) $ 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 $ p ^ {r} ( 1- qz) ^ {-} r $ in powers of $ z $.

Negative binomial distributions are encountered in many applications of probability theory. For an integer $ r > 0 $, the negative binomial distribution is interpreted as the distribution of the number of failures before the $ r $- th "success" in a scheme of Bernoulli trials with probability of "success" $ p $; in this context it is usually called a Pascal distribution and is a discrete analogue of the gamma-distribution. When $ r= 1 $, 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 $ Y $ is a random variable having, conditionally on $ \lambda $, a Poisson distribution with random parameter $ \lambda $, which in turn has a gamma-distribution with density

$$ \frac{1}{\Gamma ( \mu ) } x ^ {\mu - 1 } e ^ {- \alpha x } ,\ \ x > 0,\ \mu > 0, $$

then the marginal distribution of $ Y $ will be a negative binomial distribution with parameters $ r = \mu $ and $ p = \alpha /( 1+ \alpha ) $. The negative binomial distribution serves as a limiting form of a Pólya distribution.

The sum of independent random variables $ X _ {1} \dots X _ {n} $ which have negative binomial distributions with parameters $ p $ and $ r _ {1} \dots r _ {n} $, respectively, has a negative binomial distribution with parameters $ p $ and $ r _ {1} + \dots + r _ {n} $. For large $ r $ and small $ q $, where $ rq \sim \lambda $, the negative binomial distribution is approximated by the Poisson distribution with parameter $ \lambda $. 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)