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.
$\blacksquare$