Expectation of Bernoulli Distribution/Proof 3

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

Then the expectation of $X$ is given by:
 * $E \left({X}\right) = p$

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


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

where $q = 1 - p$.

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


 * $E \left({X}\right) = \Pi'_X \left({1}\right)$

From Derivatives of PGF of Bernoulli Distribution, we have $\Pi'_X \left({s}\right) = p$.

Hence the result.