Negative Binomial Distribution as Generalized Geometric Distribution/Second Form

Theorem
The second form of the negative binomial distribution is a generalization of the shifted geometric distribution:

Let $\sequence {Y_i}$ be a Bernoulli process with parameter $p$.

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

Proof
Consider the experiment $\FF$ as described.

By Bernoulli Process as a Negative Binomial Distribution: Second Form, $\FF$ is modelled by a negative binomial distribution of the second form with parameters $n$ and $p$:
 * $\ds \forall k \in \Z, k \ge n: \map \Pr {Y = k} = \binom {k - 1} {n - 1} q^{k - n} p^n$

where $q = 1 - p$.

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

By Bernoulli Process as Shifted Geometric Distribution, $\FF'$ is modelled by a geometric distribution with parameter $p$:
 * $\forall k \in \Z, k \ge 1: \map \Pr {Y = k} = q^{k - 1} p$

where $q = 1 - p$.