# Category:Definitions/Negative Binomial Distribution

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This category contains definitions related to Negative Binomial Distribution.

Related results can be found in Category:Negative Binomial Distribution.

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

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:

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

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

where $0 < p < 1$.

It is frequently seen as:

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

where $q = 1 - p$.

### Second Form

$X$ 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 $q = 1 - p$.

## Pages in category "Definitions/Negative Binomial Distribution"

The following 4 pages are in this category, out of 4 total.