Probability Generating Function of Negative Binomial Distribution/Second Form

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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 $q = 1 - p$.


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 $q = 1 - p$.


\(\ds \map {\Pi_X} s\) \(=\) \(\ds \sum_{k \mathop \ge n} \binom {k - 1} {n - 1} p^n q^{k - n} s^k\)
\(\ds \) \(=\) \(\ds \frac {p^n} {q^n} \sum_{k \mathop \ge n} \binom {k - 1} {n - 1} \paren {q s}^k\)
\(\ds \) \(=\) \(\ds \paren {\frac {p s} {1 - q s} }^n\)

Hence the result.