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:


 * $\Img X = \set {n, n + 1, n + 2, \dotsc}$


 * $\map \Pr {X = k} = \dbinom {k - 1} {n - 1} p^n \paren {1 - p}^{k - n}$

where $0 < p < 1$.

It is frequently seen as:
 * $\map \Pr {X = k} = \dbinom {k - 1} {n - 1} q^{k - n} p^n $

where $q = 1 - p$.

It is sometimes written:
 * $X \sim \NegativeBinomial n p$

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