# Bernoulli Process as Negative Binomial Distribution

## Theorem

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

### First Form

Let $\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a total of $n$ failures have been encountered.

Let $X$ be the discrete random variable defining the number of successes before $n$ failures have been encountered.

Then $X$ is modeled by a negative binomial distribution of the first form.

### Second Form

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 $n$ successes, and then stops.

Let $Y$ be the discrete random variable defining the number of trials before $n$ successes have been achieved.

Then $X$ is modeled by a negative binomial distribution of the second form.