Negative Binomial Distribution Gives Rise to Probability Mass Function/First Form
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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:
\(\ds \map \Pr \Omega\) | \(=\) | \(\ds \sum_{k \mathop \ge n} \binom {n + k - 1} {n - 1} p^k \paren {1 - p}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - p}^n \sum_{k \mathop \ge n} \binom {n + k - 1} k p^k\) | Symmetry Rule for Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - p}^n \sum_{j \mathop \ge 0} \binom {-n} k p^k\) | Negated Upper Index of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - p}^n p^{-n}\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
So $X$ satisfies $\map \Pr \Omega = 1$, and hence the result.
$\blacksquare$