# Negative Binomial Distribution as Generalized Geometric Distribution/First Form

## Theorem

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.

## Proof

Consider the experiment $\EE$ as described.

By Bernoulli Process as a Negative Binomial Distribution: First Form, $\EE$ is modelled by a negative binomial distribution of the first form with parameters $n$ and $p$:

- $\forall k \in \Z, k \ge 0: \map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k q^n $

where $q = 1 - p$.

Now consider the experiment $\EE'$ as described.

By Bernoulli Process as Geometric Distribution, $\EE'$ is modelled by a geometric distribution with parameter $p$:

- $\forall k \in \Z, k \ge 0: \map \Pr {X = k} = p^k q$

where $q = 1 - p$.

This theorem requires a proof.In particular: Requires further resultsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |