Negative Binomial Distribution Gives Rise to Probability Mass Function/Second 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 (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
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples