Moments of a Distribution

Moments are quantitative measures of a distribution function. Formally, the nth moment about a value c of a distribution f(x) is defined as

$$\begin{array}{l}{\mu }_n=E[(x-c{)}^n]=\\ \left\{\begin{array}{c}{\displaystyle \sum {(x-c)}^{n}f(x)\text{Discrete distibution}}\\ {\displaystyle \int {(x-c)}^{n}f(x)}dx\text{Continuous distribution}\end{array}\right\}.\end{array}$$

[Page 1084]When c = 0, they are called the raw moments, and when c is set at the mean of the distributions, they are called central moments. The first raw moment is the mean and the first central moment is 0. For the second and higher moments, the central moments are often used. For some distributions, their moments can be flexibly obtained through their moment-generating functions. Certain distributions can be uniquely determined by a few moments. For example, a normal distribution can be determined by its first two moments. Although higher moments of a distribution can be available, the first four moments are ...