Definition:Negative Binomial Distribution/Type 1

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Definition

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.


Also defined as

The type $1$ negative binomial distribution can also be defined as follows:

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 $r$ failures, and then stops.
Let $X$ be the discrete random variable defining the number of Bernoulli trials before the $r$th failure has occurred.


Then $X$ has the type $1$ negative binomial 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.


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.


Also see

  • Results about the type 1 negative binomial distribution 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


Sources

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