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

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:

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