Moments

Definition

In general, if $m$ is a positive integer, and if $M^{(m)}(t)$ means the $m$-th derivative of $M(t)$, we have, by repeated differentiation with respect to $t$, $M^{(m)}(0)=E(X^m)$

Now $E(X^m)={\int }_{-\infty }^{\infty }{x}^{m}f(x)dx\text{or}{\sum }_x{x}^{m}p(x),$ and in mechanics, the integrals (or sums) of this sort are called moments.

Remark

Since $M(t)$ generates the values of $E(X^m),m=1,2,3,\dots$, it is called the moment generating function (mgf).

We sometimes call $E(X^m)$ the $m$-th moment of the distribution, or the $m$-th moment of $X$.

Some important properties of moments include:

$$\begin{array}{rl}E(X)& =\mu \\ E(X^2)& =\mathrm{Var}(X)+[E(X){]}^2\end{array}$$

For a random sample $X_1,X_2,\dots ,X_n$, the $m$-th sample moment is defined as: $M_m=\frac{1}{n}{\sum }_{i=1}^n{X}_i^m$

The first few sample moments are:

• First sample moment (sample mean): $M_1=\bar{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$
• Second sample moment: $M_2=\frac{1}{n}{\sum }_{i=1}^n{X}_i^2$
• Third sample moment: $M_3=\frac{1}{n}{\sum }_{i=1}^n{X}_i^3$