Negative Binomial Distribution Gives Rise to Probability Mass Function

Theorem

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

First Form

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.

Second Form

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

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