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

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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.


Proof

By definition:

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


Then:

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


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

$\blacksquare$


Sources