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Definition (Mathematical Expectation)

For any function deterministic function $g(x)$, the expected value of $g(X),$ which is denoted $E[g(X)]$, is defined as follows:

$$ E[g(X)] := \sum_{x \in \mathcal{X}} g(x) p(x).

$$

In the special case where $g(x) = x,$ we call $E[X]$ the mean of $X.$

The variance of $X$, which is denoted $\text{Var}(X),$ is defined by the relationship

$$ \text{Var}(X) := E\big[(X - E[X])^2\big] $$

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Expected Value (Mean) of a Random Variable and Standard Deviation

The expected value of a discrete random variable is written

$$ E(X) = \sum_{x \in \mathcal{X}} p(x) $$

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Until I am able to place some content here, see any of the following pages for some quick examples and an overview.