Definition:Negative Binomial Distribution/First Form

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


 * $\Img X = \set {0, 1, 2, \ldots}$


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

where $0 < p < 1$.

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

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 (First Form) as Generalized Geometric Distribution‎


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