Negative Binomial Distribution as Generalized Geometric Distribution/Second Form

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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 $q = 1 - p$.