Expectation

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

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

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 $\mathbb E[X]=\int_0^1 x\,dx=\tfrac12$.