I have three random variables $X,Y,Z$ that all have a mean of $\mu=10$ and a standard deviation of $\sigma=2$. Now, I would like to say that they are negatively correlated with $\rho_{X,Y}=-1, \rho_{Y,Z}=-1, \rho_{X,Z}=-1$.
My question would be, what is the variance of $X+Y+Z$? Normally, we would say: $Var(X+Y+Z)=Var(X)+Var(X)+Var(Z)+2Cov(X,Y)+2Cov(Y,Z)+2Cov(X,Z)$
However, in our simple example, this leads to $3\cdot2^2+3\cdot2\cdot(-1)\sqrt{2^2\cdot2^2}=-12$ which cannot make sense.
Where is my logical error? I presume because of some transitivity, $\rho_{X,Z}$ cannot be $-1$ if the other two $\rho_i$ are -1 but which principle is behind this?