Definition:Negative Binomial Distribution

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

Then $$X$$ has the negative binomial distribution 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 $$0 < p < 1$$.

Note that the Negative Binomial Distribution Gives Rise to Probability Mass Function satisfying $$\Pr \left({\Omega}\right) = 1$$.