Definition:Negative Binomial Distribution/Second Form

Definition
Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. $X$ has the negative binomial distribution (of the second form) with parameters $n$ and $p$ if:


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


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

where $0 < p < 1$.

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

where $q = 1 - p$.

It is sometimes written:
 * $X \sim \operatorname{NB} \left({n, p}\right)$

but there is no standard notation for this distribution.

Also see

 * Negative Binomial Distribution (Second Form) as Generalized Geometric Distribution


 * Negative Binomial Distribution (Second Form) Gives Rise to Probability Mass Function