Probability Generating Function of Negative Binomial Distribution/Second Form
Jump to navigation
Jump to search
Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.
Then the p.g.f. of $X$ is:
- $\ds \map {\Pi_X} s = \paren {\frac {p s} {1 - q s} }^n$
where $q = 1 - p$.
Proof
From the definition of p.g.f:
- $\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map {p_X} k s^k$
From the definition of the negative binomial distribution (second form):
- $\map {p_X} k = \dbinom {k - 1} {n - 1} p^n q^{k - n}$
where $q = 1 - p$.
So:
\(\ds \map {\Pi_X} s\) | \(=\) | \(\ds \sum_{k \mathop \ge n} \binom {k - 1} {n - 1} p^n q^{k - n} s^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {p^n} {q^n} \sum_{k \mathop \ge n} \binom {k - 1} {n - 1} \paren {q s}^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^n\) |
Hence the result.
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: $(13)$