Expectation of Bernoulli Distribution/Proof 1

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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 definition of expectation:

$\displaystyle \expect X = \sum_{x \mathop \in \Img X} x \, \map \Pr {X = x}$

By definition of Bernoulli distribution:

$\expect X = 1 \times p + 0 \times \paren {1 - p}$

Hence the result.

$\blacksquare$