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:


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

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:

$$ $$

Hence the result.