25
Oct 16

## Properties of variance

### All properties of variance in one place

Certainty is the mother of quiet and repose, and uncertainty the cause of variance and contentions. Edward Coke

Preliminaries: study properties of means with proofs.

Definition. Yes, uncertainty leads to variance, and we measure it by $Var(X)=E(X-EX)^2$. It is useful to use the name deviation from mean for $X-EX$ and realize that $E(X-EX)=0$, so that the mean of the deviation from mean cannot serve as a measure of variation of $X$ around $EX$.

Property 1. Variance of a linear combination. For any random variables $X,Y$ and numbers $a,b$ one has
(1) $Var(aX + bY)=a^2Var(X)+2abCov(X,Y)+b^2Var(Y).$
The term $2abCov(X,Y)$ in (1) is called an interaction term. See this post for the definition and properties of covariance.
Proof.
$Var(aX + bY)=E[aX + bY -E(aX + bY)]^2$

(using linearity of means)
$=E(aX + bY-aEX -bEY)^2$

(grouping by variable)
$=E[a(X-EX)+b(Y-EY)]^2$

(squaring out)
$=E[a^2(X-EX)^2+2ab(X-EX)(Y-EY)+(Y-EY)^2]$

(using linearity of means and definitions of variance and covariance)
$=a^2Var(X) + 2abCov(X,Y) +b^2Var(Y).$
Property 2. Variance of a sum. Letting in (1) $a=b=1$ we obtain
$Var(X + Y) = Var(X) + 2Cov(X,Y)+Var(Y).$

Property 3. Homogeneity of degree 2. Choose $b=0$ in (1) to get
$Var(aX)=a^2Var(X).$
Exercise. What do you think is larger: $Var(X+Y)$ or $Var(X-Y)$?
Property 4. If we add a constant to a variable, its variance does not change: $Var(X+c)=E[X+c-E(X+c)]^2=E(X+c-EX-c)^2=E(X-EX)^2=Var(X)$
Property 5. Variance of a constant is zero: $Var(c)=E(c-Ec)^2=0$.

Property 6. Nonnegativity. Since the squared deviation from mean $(X-EX)^2$ is nonnegative, its expectation is nonnegative$E(X-EX)^2\ge 0$.

Property 7. Only a constant can have variance equal to zero: If $Var(X)=0$, then $E(X-EX)^2 =(x_1-EX)^2p_1 +...+(x_n-EX)^2p_n=0$, see the definition of the expected value. Since all probabilities are positive, we conclude that $x_i=EX$ for all $i$, which means that $X$ is identically constant.

Property 8. Shortcut for variance. We have an identity $E(X-EX)^2=EX^2-(EX)^2$. Indeed, squaring out gives

$E(X-EX)^2 =E(X^2-2XEX+(EX)^2)$

(distributing expectation)

$=EX^2-2E(XEX)+E(EX)^2$

(expectation of a constant is constant)

$=EX^2-2(EX)^2+(EX)^2=EX^2-(EX)^2$.

All of the above properties apply to any random variables. The next one is an exception in the sense that it applies only to uncorrelated variables.

Property 9. If variables are uncorrelated, that is $Cov(X,Y)=0$, then from (1) we have $Var(aX + bY)=a^2Var(X)+b^2Var(Y).$ In particular, letting $a=b=1$, we get additivity$Var(X+Y)=Var(X)+Var(Y).$ Recall that the expected value is always additive.

Generalizations$Var(\sum a_iX_i)=\sum a_i^2Var(X_i)$ and $Var(\sum X_i)=\sum Var(X_i)$ if all $X_i$ are uncorrelated.

Among my posts, where properties of variance are used, I counted 12 so far.