Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$.

If $X\ge 0$ pointwise, the \textbf{expectation} of $X$ is $\mathbb{E}[X]=\int_\Omega X\,dP\in[0,\infty]$, the integral of \ref{def:lebesgue-integral-nonnegative-2026a}. If $X$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $P$, then $\mathbb{E}[X]=\int_\Omega X\,dP\in\mathbb{R}$ and $X$ is said to have \textbf{finite expectation}.

For $k\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, the power $X^{k}$ is a random variable: for $a\ge 0$, $\{X^{k}>a\}$ equals $\{X>a^{1/k}\}$ for odd $k$ and $\{X>a^{1/k}\}\cup\{X<-a^{1/k}\}$ for even $k$ (roots exist by \ref{thm:nth-root-rudin-b}); for $a<0$, it equals $\Omega$ for even $k$ and $\{X>-|a|^{1/k}\}$ for odd $k$; in all cases this is an event by the criterion of \ref{def:measurable-function-2026a}. If $X^{k}$ has finite expectation, $\mathbb{E}[X^{k}]$ is the \textbf{$k$th moment} of $X$.

If $X$ and $X^{2}$ have finite expectation, the \textbf{variance} of $X$ is

which is finite and equals $\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}$ by \ref{thm:linearity-monotonicity-integral-2026a}.