Negative Binomial Distribution as Generalized Geometric Distribution

Theorem

First Form

The first form of the negative binomial distribution is a generalization of the geometric distribution:

Let $\sequence {X_i}$ be a Bernoulli process with parameter $p$.

Let $\EE$ be the experiment which consists of:

Perform the Bernoulli trial $X_i$ until $n$ failures occur, and then stop.

Let $k$ be the number of successes before before $n$ failures have been encountered.

Let $\EE'$ be the experiment which consists of:

Perform the Bernoulli trial $X_i$ until one failure occurs, and then stop.

Then $k$ is modelled by the experiment:

Perform experiment $\EE'$ until $n$ failures occur, and then stop.

Second Form

The second form of the negative binomial distribution is a generalization of the shifted geometric distribution:

Let $\sequence {Y_i}$ be a Bernoulli process with parameter $p$.

Let $\FF$ be the experiment which consists of:

Perform the Bernoulli trial $Y_i$ as many times as it takes to achieve $n$ successes, and then stop.

Let $k$ be the number of Bernoulli trials that need to be taken in order to achieve up to (and including) the $n$th success.

Let $\FF'$ be the experiment which consists of:

Perform the Bernoulli trial $Y_i$ until one success is achieved, and then stop.

Then $k$ is modelled by the experiment:

Perform experiment $\FF'$ until $n$ failures occur, and then stop.