Moment

The $n$-th moment of a real-valued continuous function $f(x)$ of a real variable about a value $c$ is

$${\mu }_n={\int }_{-\infty }^{\infty }(x-c{)}^{n}f(x)\mathrm{d}x$$

Restricted to the context of Random Variables, we consider a probability measure $P$. Then the $n$-th moment of a random variable $X$ is

$$\mathbb{E}[X^n]={\int }_{-\infty }^{\infty }{x}^n\mathrm{d}P(x).$$

The $n$-th central moment is $\mathbb{E}[(X-\mathbb{E}[X]{)}^n]$. Mean is the first moment, and Variance is the second central moment.

p-Order Integrable

A r.v. is called $p$-order integrable if $\mathbb{E}[|X{|}^p]<\infty$.

• If $p>q$, then $p$-order integrable implies $q$-order integrable.