# Negative Binomial Distribution Gives Rise to Probability Mass Function/First Form

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## Theorem

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have the negative binomial distribution (first form) with parameters $n$ and $p$ ($0 < p < 1$).

Then $X$ gives rise to a probability mass function.

## Proof

By definition:

$\Img X = \set {0, 1, 2, \ldots}$
$\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k \paren {1 - p}^n$

Then:

 $\ds \map \Pr \Omega$ $=$ $\ds \sum_{k \mathop \ge n} \binom {n + k - 1} {n - 1} p^k \paren {1 - p}^n$ $\ds$ $=$ $\ds \paren {1 - p}^n \sum_{k \mathop \ge n} \binom {n + k - 1} k p^k$ Symmetry Rule for Binomial Coefficients $\ds$ $=$ $\ds \paren {1 - p}^n \sum_{j \mathop \ge 0} \binom {-n} k p^k$ Negated Upper Index of Binomial Coefficient $\ds$ $=$ $\ds \paren {1 - p}^n p^{-n}$ Binomial Theorem $\ds$ $=$ $\ds 1$

So $X$ satisfies $\map \Pr \Omega = 1$, and hence the result.

$\blacksquare$