Mean

Definition 11.8 (Mean). For a random variable $X$ with pmf/pdf $f(x)$, the expected value (mean), written $E(X)$ or $\mu$, is

$E(X)$ need not itself be a value $X$ can take (e.g. the mean number of heads in $2$ tosses is $1$, but the mean of $3$ coins' tails-count can be a fraction). It is best interpreted as the long-run average value of $X$ over many independent repetitions of the underlying experiment.

Theorem 11.3 (expectation of a function of $X$, without proof). For any function $g(X)$ of the random variable $X$,

$E(g(X))=\begin{cases}\displaystyle\sum_x g(x)\,f(x) & X\text{ discrete}\\[4pt] \displaystyle\int_{-\infty}^{\infty} g(x)\,f(x)\,dx & X\text{ continuous.}\end{cases}$ …