Expectation of Bernoulli Distribution/Proof 3

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Theorem

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


Then the expectation of $X$ is given by:

$\expect X = p$


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.

$\blacksquare$