Expectation (math)

The mathematical expectation of a continuously distributed random variable $X$ with probability density function $f(x)$ is

${\mbox{E}}[X]=\int \limits _{-\infty }^{\infty }xf(x)dx.$

The expectation is also known as the mean of $X$.

The expectation with respect to some function $g(X)$ where $X$ is distributed according to $f(x)$ is

${\mbox{E}}[g(X)]=\int \limits _{-\infty }^{\infty }g(x)f(x)dx.$

For a discretely distributed random variable $X$ with probability mass function $p_{k}$ it is

${\mbox{E}}[X]=\sum _{k}p_{k}x_{k}.$