Expectation, Variance, and Moments

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

If $X\ge 0$ pointwise, the expectation of $X$ is $\mathbb{E}[X]=\int_\Omega X\,dP\in[0,\infty]$, the integral of Lebesgue Integral of a Nonnegative Measurable Function. If $X$ is integrable with respect to $P$, then $\mathbb{E}[X]=\int_\Omega X\,dP\in\mathbb{R}$ and $X$ is said to have finite expectation.

For $k\in$ $\mathbb{N}$, 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 Existence and uniqueness of nth roots); 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 Measurable Function and Real-Valued Measurable Function. If $X^{k}$ has finite expectation, $\mathbb{E}[X^{k}]$ is the $k$th moment of $X$.

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

which is finite and equals $\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}$ by Linearity and Monotonicity of the Lebesgue Integral.