Probability Generating Function of Negative Binomial Distribution
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Theorem
First Form
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 $q = 1 - p$.
Second Form
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 $q = 1 - p$.