# Probability Generating Function of Binomial Distribution

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## Theorem

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 $q = 1 - p$.

## 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 binomial distribution:

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

So:

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

Hence the result.

$\blacksquare$