Moment

A moment of order $k$ is an expectation of a power of a random variable $X$, typically the raw moment $\mathbb{E}[X^k]$ or the central moment $\mathbb{E}\!\left[(X-\mathbb{E}[X])^k\right]$, whenever these expectations exist (equivalently, when $\mathbb{E}[|X|^k]<\infty$).

Moments summarize aspects of the distribution ; for instance the second central moment is the variance and the first raw moment is the mean $\mathbb{E}[X]$. When a moment generating function exists in a neighborhood of zero, its derivatives recover the raw moments.

Examples:

• If $X\sim\mathrm{Bernoulli}(p)$, then $X^k=X$ for all integers $k\ge 1$, so $\mathbb{E}[X^k]=p$ and $\operatorname{Var}(X)=p(1-p)$.
• If $X\sim N(0,1)$, then $\mathbb{E}[X]=0$, $\mathbb{E}[X^2]=1$, and all odd moments are $0$.
• If $X\sim\mathrm{Exp}(\lambda)$ with $\lambda>0$, then $\mathbb{E}[X^k]=k!/\lambda^k$ for integers $k\ge 1$.