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 $q = 1 - p$.
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