Probability Generating Function of Bernoulli Distribution

Theorem

Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$.

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

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

where $q = 1 - p$.

Proof

From the definition of p.g.f:

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

From the definition of the Bernoulli distribution:

$p_X \left({x}\right) = \begin{cases} p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \left\{{a, b}\right\} \\ \end{cases}$

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pi_X \left({s}\right)\)	\(=\)	\(\displaystyle p_X \left({0}\right) s^0 + p_X \left({1}\right) s^1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({1-p}\right) + p s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Hence the result.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 4.2 \ (10)$