Negative Binomial Distribution Gives Rise to Probability Mass Function/Second 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 with parameters $n$ and $p$ ($$0 < p < 1$$).

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

Proof
By definition:


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


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

Then:

$$ $$ $$ $$ $$

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