Probability Generating Function of Binomial Distribution

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:

$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$

where $q = 1 - p$.

Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{k \ge 0} p_X \left({k}\right) s^k$

From the definition of the binomial distribution:

$\displaystyle p_X \left({k}\right) = \binom n k p^k \left({1-p}\right)^{n-k}$

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pi_X \left({s}\right)\)	\(=\)	\(\displaystyle \sum_{k=0}^n \binom n k p^k \left({1-p}\right)^{n-k} s^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k=0}^n \binom n k \left({p s}\right)^k \left({1-p}\right)^{n-k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({p s}\right) + \left({1-p}\right)}\right)^n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by the Binomial Theorem

Hence the result.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 4.2 \ (11)$