Definition:Negative Binomial Distribution

Let $$X$$ be a discrete random variable on a probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

There are two forms of the negative binomial distribution, as follows:

First Form
$$X$$ has the negative binomial distribution (of the first form) with parameters $$n$$ and $$p$$ if:


 * $$\operatorname{Im} \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\}$$


 * $$\Pr \left({X = k}\right) = \binom {n + k - 1} {n - 1} p^k \left({1-p}\right)^n$$

where $$0 < p < 1$$.

It is frequently seen as:
 * $$\Pr \left({X = k}\right) = \binom {n + k - 1} {n - 1} p^k q^n $$

where $$q = 1 - p$$.

This is a generalization of the geometric distribution.

That is, it can be viewed as modelling the number of successes in a series of Bernoulli trials before $$n$$ failures have been encountered.

Second Form
$$X$$ has the negative binomial distribution (of the second form) with parameters $$n$$ and $$p$$ if:


 * $$\operatorname{Im} \left({X}\right) = \left\{{n, n+1, n+2, \ldots}\right\}$$


 * $$\Pr \left({X = k}\right) = \binom {k-1} {n-1} p^n \left({1-p}\right)^{k-n}$$

where $$0 < p < 1$$.

It is frequently seen as:
 * $$\Pr \left({X = k}\right) = \binom {k-1} {n-1} q^{k-n} p^n $$

where $$q = 1 - p$$.

This is a generalization of the shifted geometric distribution.

That is, it can be viewed as modelling the number of Bernoulli trials up to (and including) the $$n$$th success.