Three random variables equation

Question

Given the random variables X,Y,Z have the same distribution and fulfil the following equalities: $$Var(X+Y+Z)=21,Cov(X,Y)=Cov(Y,Z)=Cov(Z,X)=1$$ Find $VarX$ and $Var(X+Y)$. I am lost with this question how it should be approached?

Answer 1

$$Var(X+Y+Z)=Var(X)+Var(Y)+Var(Z)+2Cov(X,Y)+2Cov(X,Z)+2Cov(Y,Z)$$ And since X, Y, and Z have the same distribution $$Var(X+Y+Z)=3Var(X)+2Cov(X,Y)+2Cov(X,Z)+2Cov(Y,Z)$$

Answer 2

Hints:

• $\text{Var}(U)=\text{Cov}(U,U)$.
• $\text{Cov}$ is symmetric and bilinear.