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 $p$:
 * $$\forall k \in \Z, k \ge 0: \Pr \left({X = k}\right) = \binom {n + k - 1} {n - 1} p^k q^n $$

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 $p$:
 * $$\forall k \in \Z, k \ge n: \Pr \left({Y = k}\right) = \binom {k-1} {n-1} q^{k-n} p^n $$

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 $$q = 1 - p$$.