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:


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

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:

Hence the result.