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:
Type $1$
Let $\sequence {X_i}$ be a Bernoulli process with parameter $p$.
Let $\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ as many times as it takes to achieve a total of $r$ successes, and then stops.
Let $X$ be the discrete random variable defining the number of Bernoulli trials before the $r$th success has occurred.
Then $X$ has the type $1$ negative binomial distribution.
Type $2$
Let $\sequence {X_i}$ be a Bernoulli process with parameter $p$.
Let $\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a total of $r$ failures have occurred.
Let $X$ be the discrete random variable defining the number of successes before $r$ failures have occurred.
Then $X$ has a type 2 negative binomial distribution.
Notation
A negative binomial distribution (either type) can be written:
- $X \sim \NegativeBinomial r p$
There appears to be no standard notation for this distribution.
Terminology
There are two ways the negative binomial distribution is defined in the literature, both based on a Bernoulli Process:
- Type $(1)$: the total number of Bernoulli trials before the $r$th success (or failure) has occurred
- Type $(2)$: the total number of failures (or successes) before the $r$th success (or failure) has occurred.
Most sources give only one of these types of negative binomial distribution, defining it as either one or the other.
$\mathsf{Pr} \infty \mathsf{fWiki}$ is careful to define both types, and to distinguish between the two of them as appropriate.
Also see
- Results about negative binomial distributions can be found here.
Technical Note
The $\LaTeX$ code for \(\NegativeBinomial {r} {p}\) is \NegativeBinomial {r} {p} .
When the arguments are single characters, it is usual to omit the braces:
\NegativeBinomial r p