Bernoulli Process as Negative Binomial Distribution
Jump to navigation
Jump to search
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.