Expectation of Bernoulli Distribution

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 1
From the definition of expectation:
 * $E \left({X}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x \Pr \left({X = x}\right)$

By definition of Bernoulli distribution:
 * $E \left({X}\right) = 1 \times p + 0 \times \left({1-p}\right)$

Hence the result.

Proof 2
We can also use the Expectation of Binomial Distribution putting $n = 1$.

Proof 3
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 P.G.F., 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.