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

Theorem
Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. 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:


 * $\operatorname{Im} \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\}$


 * $\displaystyle \Pr \left({X = k}\right) = \binom {n + k - 1} {n - 1} p^k \left({1-p}\right)^n$

Then:

So $X$ satisfies $\Pr \left({\Omega}\right) = 1$, and hence the result.