Expectation of Bernoulli Distribution/Proof 3

Proof
From the Probability Generating Function of Bernoulli Distribution, we have:


 * $\map {\Pi_X} s = q + p s$

where $q = 1 - p$.

From Expectation of Discrete Random Variable from PGF, we have:


 * $\expect X = \map { {\Pi_X}'} 1$

From Derivatives of PGF of Bernoulli Distribution:
 * $\map { {\Pi_X}'} s = p$

Hence the result.