Definition:Negative Binomial Distribution/First Form
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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 $q = 1 - p$.
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/First Form
- Negative Binomial Distribution (First Form) as Generalized Geometric Distribution‎
- Negative Binomial Distribution (First Form) 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