Bernoulli Process as Negative Binomial Distribution

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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.