Covariance of 3 variables

Question

$\newcommand{Cov}{\operatorname{Cov}}X,Y,Z$ are three random variables each with mean $0$ and variance $20$. $\Cov(X,Y)=\Cov(X,Z)=10$ and $\Cov(Y,Z)=5$. What is $\Cov(3X+Z,3X+Y)$?

Answer 1

$\newcommand{Cov}{\operatorname{Cov}}$Covariance behaves like multiplication in that it distributes over linear combinations of random variables. Thus $$\Cov(3X+Z,3X+Y)=9\Cov(X,X)+3\Cov(X,Y)+3\Cov(X,Z)+\Cov(Y,Z)=9\operatorname{Var}(X)+3\cdot10+3\cdot10+5=180+30+30+5=245$$