# Definition:Negative Binomial Distribution

## Definition

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 $q = 1 - p$.

## Notation

The **negative binomial distribution** (in either form) can be written:

- $X \sim \NegativeBinomial n p$

but there is no standard notation for this distribution.

## Also see

- Bernoulli Process as Negative Binomial Distribution
- Negative Binomial Distribution Gives Rise to Probability Mass Function

- Results about
**the negative binomial distribution**can be found here.

## Technical Note

The $\LaTeX$ code for \(\NegativeBinomial {n} {p}\) is `\NegativeBinomial {n} {p}`

.

When the arguments are single characters, it is usual to omit the braces:

`\NegativeBinomial n p`