Probability Generating Function of Negative Binomial Distribution/First Form
< Probability Generating Function of Negative Binomial Distribution(Redirected from Probability Generating Function of Negative Binomial Distribution (First Form))
Jump to navigation
Jump to search
Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$.
Then the p.g.f. of $X$ is:
- $\map {\Pi_X} s = \paren {\dfrac q {1 - p s} }^n$
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 (first form):
- $\map {p_X} k = \dbinom {n + k - 1} {n - 1} p^k q^n$
where $q = 1 - p$.
So:
\(\ds \map {\Pi_X} s\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom {n + k - 1} {n - 1} p^k q^n s^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds q^n \sum_{k \mathop \ge 0} \binom {n + k - 1} {n - 1} \paren {p s}^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac q {1 - p s} }^n\) |
For the third equality, the equation in the second line is rewritten in terms of binomial series.
This article, or a section of it, needs explaining. In particular: Yes all very well, but it's still not trivial. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Hence the result.
$\blacksquare$