Variance

Definition

Let

• $X$ : Random variable, with
  • Finite mean $\mu$
  • Finite $E[(X-\mu {)}^2]$

If

$${\sigma }^2=\mathrm{Var}(X)=E[(X-\mu {)}^2]$$

Then we say ${\sigma }^2$ is the variance of $X$

Remark

The variance is often referred to as the second moment (about $\mu$) of a random variable.

An important equation on the second moment (about 0) can be obtained from the variance and mean of $X$.

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

Related theorems

• Constant multiplication and addition with vairance