# 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)