Expectation (math)

The mathematical expectation of a continuously distributed random variable <math>X</math> with probability density function <math>f(x)</math> is

<math>

\mbox{E}[X] =\int\limits_{-\infty}^\infty x f(x)dx. </math> The expectation is also known as the mean of <math>X</math>.

The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is

<math>

\mbox{E}[g(X)] =\int\limits_{-\infty}^\infty g(x) f(x)dx. </math>

For a discretely distributed random variable <math>X</math> with probability mass function <math>p_{k}</math> it is

<math>

\mbox{E}[X]=\sum_{k} p_{k}x_{k}. </math>