Negative Binomial Distribution as Generalized Geometric Distribution

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

Let $\left \langle{X_i}\right \rangle$ be a Bernoulli process with parameter $p$.

Let $\mathcal E$ 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 $\mathcal E'$ 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 $\mathcal E'$ 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 $\left \langle{Y_i}\right \rangle$ be a Bernoulli process with parameter $p$.

Let $\mathcal F$ 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 $\mathcal F'$ 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 $\mathcal F'$ until $n$ failures occur, and then stop.

Proof of First Form
Consider the experiment $\mathcal E$ as described.

By Bernoulli Process as a Negative Binomial Distribution: First Form, $\mathcal E$ is modelled by a negative binomial distribution of the first form with parameters $n$ and $p$:
 * $\displaystyle \forall k \in \Z, k \ge 0: \Pr \left({X = k}\right) = \binom {n + k - 1} {n - 1} p^k q^n $

where $q = 1 - p$.

Now consider the experiment $\mathcal E'$ as described.

By Bernoulli Process as a Geometric Distribution, $\mathcal E'$ is modelled by a geometric distribution with parameter $p$:
 * $\forall k \in \Z, k \ge 0: \Pr \left({X = k}\right) = p^k q$

where $q = 1 - p$.

Proof of Second Form
Consider the experiment $\mathcal F$ as described.

By Bernoulli Process as a Negative Binomial Distribution: First Form, $\mathcal F$ is modelled by a negative binomial distribution of the second form with parameters $n$ and $p$:
 * $\displaystyle \forall k \in \Z, k \ge n: \Pr \left({Y = k}\right) = \binom {k-1} {n-1} q^{k-n} p^n$

where $q = 1 - p$.

Now consider the experiment $\mathcal F'$ as described.

By Bernoulli Process as a Geometric Distribution, $\mathcal F'$ is modelled by a geometric distribution with parameter $p$:
 * $\forall k \in \Z, k \ge 1: \Pr \left({Y = k}\right) = q^{k-1} p$

where $q = 1 - p$.