Variance

Definition

The variance of a r.v. $X$ is

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

The square root of the variance is called the standard deviation (SD)

$$\mathrm{SD}(X)=\sqrt{\mathrm{Var}(X)}$$

With the linearity of expectation, we have

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

Since $\mathrm{Var}(X)\ge 0$, we then have $\mathrm{E}\left[X^2\right]\ge {\mathrm{E}}^2[X]$

Property

• $\mathrm{Var}(X+c)=\mathrm{Var}(X)$ for any constant $c$
• $\mathrm{Var}(cX)=c^2\mathrm{Var}(X)$ for any constant $c$
• If $X$ and $Y$ are independent, then $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$

Note

Attention: $\mathrm{Var}(X-Y)=\mathrm{Var}(X)+\mathrm{Var}(-Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$

• For any constant $c$, $\mathrm{Var}(X)\le \mathrm{E}\left[{\left(X-c\right)}^2\right]$. The equality holds if and only if $c=\mathrm{E}[X]$. This means $\mathrm{E}\left[(X-c{)}^2\right]$ reaches its minimum when $c=\mathrm{E}[X]$, and the value of this minimum is $\mathrm{Var}(X)$