Moments

Definition

$r$-th moment is defined as

$$m_r=E[X^r]$$

Definition

$r$-th central moment is defined as

$${\mu }_r=E[(X-\mu {)}^2]$$

Mean and variance

First moment is the $m_1=\mu$ mean. And the first central moment is $0$.

$$m_1=\mu =E[X^1]$$ $${\mu }_1=E[X-\mu ]=E[X]-E[\mu ]=\mu -\mu =0$$

Second central moment is variance

$${\mu }_2=E[(X-\mu {)}^2]=E[X^2]-E[X{]}^2=m_2-m_1^2$$

then

$$m_2=E[X^2]={\mu }_2+m_1^2={\sigma }^2+{\mu }^2$$

Note that $V(X)=0$ if $X$ is degenerate

Recursive calculation

If one knows $m_1,m_2\dots m_k$ then ${\mu }_1,{\mu }_2\dots {\mu }_k$ can be computed

$${\mu }_k=E[(X-\mu {)}^k]=m_k-\left(\begin{array}{c}k\\ 1\end{array}\right)\mu m_{k-1}+\left(\begin{array}{c}k\\ 2\end{array}\right){\mu }^2{m}_{k-2}\cdots +(-1{)}^k{\mu }^k$$

and vice versa

$$m_k=E[(X-\mu +\mu {)}^k]={\mu }_k+\left(\begin{array}{c}k\\ 1\end{array}\right)\mu {\mu }_{k-1}+\left(\begin{array}{c}k\\ 2\end{array}\right){\mu }^2{\mu }_{k-2}\cdots +{\mu }^k$$

Remarks

1. For some random variables moments up to some order exists.
2. If the moment of order $k$ exists for a rv then $m_s$ exists for all $0\le s\le k$.