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