If $X$ and $Y$ are negatively correlated and $Y,Z$ are negatively correlated, does that mean $X,Z$ are positively correlated?

Question

If $Cov(X,Y)<0$ and $Cov(Y,Z)<0$ does that necessarily mean $Cov(X,Z)>0$? Intuitively I'm thinking yes, but I'm trying to prove it. My thoughts were to look at $E(XZ)$ and condition that on $X$ to see if that helps somehow. I was also thinking that you could consider covariance to be like the angle between two random variables and relate that to cosine somehow but in this case we're not assuming the random variables are centered either.

Answer 1

Let $X,W$ be independent variables with mean $0$ and variance $1$. Define $Y= W-X$ and $Z=-2W-X$.

Then Cov$(X,Y)=-1$, $\,$ Cov$(X,Z)=-1$ and Cov$(Y,Z)=-1$.