Probability Generating Function of Binomial Distribution

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Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:

$\map {\Pi_X} s = \paren {q + p s}^n$

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 binomial distribution:

$\map {p_X} k = \dbinom n k p^k \paren {1 - p}^{n - k}$


\(\ds \map {\Pi_X} s\) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k p^k \paren {1 - p}^{n - k} s^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k \paren {p s}^k \paren {1 - p}^{n - k}\)
\(\ds \) \(=\) \(\ds \paren {\paren {p s} + \paren {1 - p} }^n\) Binomial Theorem

Hence the result.