Moment - Knowlpedia

For a nonnegative integer $k$ , 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 $\mathbb{E}[|X|^k]<\infty$ ).

Remarks

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 $k\ge 1$ .