Useful Properties of Variance and Covariance

Variance and covariance have several useful algebraic properties which appear repeatedly in probability, statistics and econometrics.

For a linear transformation of a random variable,

$$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X).$$

Adding a constant does not change the spread of a random variable, while multiplying by $a$ multiplies its variance by $a^2$.

For two random variables $X$ and $Y$,

$$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y).$$

If $X$ and $Y$ are independent, their covariance is zero, giving the particularly useful result

$$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y).$$

Covariance generalises variance, since

$$\operatorname{Cov}(X,X)=\operatorname{Var}(X).$$

It is also linear in each of its two arguments. For example,

$$\operatorname{Cov}(aX+b,Y)=a\operatorname{Cov}(X,Y).$$

More generally,

$$\operatorname{Cov}(aU+bV+c,dX+eY+f)=ad\operatorname{Cov}(U,X)+ae\operatorname{Cov}(U,Y)+bd\operatorname{Cov}(V,X)+be\operatorname{Cov}(V,Y).$$

These properties make variance and covariance particularly convenient to work with algebraically, and they are used frequently in econometrics.

Background:

• Variance and Standard Deviation

• Covariance and Correlation