Expectation (math)

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

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

For a discretely distributed variable $X$ with probability mass function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{k}} it is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{E}[X] = \sum_{k} p_{k}x_{k} }

The expectation is also the mean of a distributed variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

The expectation with respect to some function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(X)} where $X$ is distributed according to $f(x)$ is

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