Category:Definitions/Negative Binomial Distributions

This category contains definitions related to the negative binomial distribution.
Related results can be found in Category:Negative Binomial Distributions.

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.