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:
- $\ds \expect X = \sum_{x \mathop \in \Img X} x \map \Pr {X = x}$
By definition of Bernoulli distribution:
- $\expect X = 1 \times p + 0 \times \paren {1 - p}$
Hence the result.
$\blacksquare$