Expectation - Imported Knowlpedia Content

An expectation of a random variable $X$ is the number

\mathbb E[X]=\int_\Omega X(\omega)\,d\mathbb P(\omega),

provided $X$ is integrable, i.e. $\int_\Omega |X|\,d\mathbb P<\infty$ (so $X$ is an $L^1$ random variable; see L1 function).

Remarks

This definition uses the Lebesgue integral on the underlying probability space; expectation is the basic averaging operation underlying variance, covariance, and many limit theorems.

Examples

If $X$ takes values $x_k$ with probabilities $p_k$ (countably many), then $\mathbb E[X]=\sum_k x_k p_k$ whenever $\sum_k |x_k|p_k<\infty$ .
If $X$ is uniform on $[0,1]$ , then $\mathbb E[X]=\int_0^1 x\,dx=\tfrac12$ .