In this post we are going to go over the derivation of probability generating function of the Bernoulli distribution. Probability Generating Function The probability generating function often referred to as the PGF has the following definition $$ \mathcal{G}_{X} (z) = \mathbb{E}_{X}[z^k] = \sum_{k = 0}^{\infty} P(x = k)z^k $$ Bernoulli As can be recalled from previous posts, the Bernoulli distribution has the PDF $$ f(x) = {n\choose k} p^x (1- p)^x $$