Probability Generating Function of Binomial Distribution

Theorem

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:

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

So:

$\ds \map {\Pi_X} s$

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k p^k \paren {1 - p}^{n - k} s^k$

$\ds $

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k \paren {p s}^k \paren {1 - p}^{n - k}$

$\ds $

$=$

$\ds \paren {\paren {p s} + \paren {1 - p} }^n$

Hence the result.

$\blacksquare$