If I have two random variables $X,Y$ , $X>0$ when $Y<0$ and $X<0$ when $Y>0$ then $\operatorname{cov}(X,Y)<0$?

Question

So as I said in the title- I'm talking about the covariance of $X$ and $Y$.

Also, is it correct to say that if I know that $Y>0$ iff $X>0$, and also $Y<0$ iff $X<0$ then $\operatorname{cov}(X,Y)>0$?

It makes sense to me but I just can't see it by the definition of covariance.

Answer 1

This is certainly true when the expected values of $X$ and $Y$ are both $0$, as is easily verified.

Now consider $\{(-1,-1)\} \cup \{ (j,1000-j) : j=1,\ldots,1000\}.$ Software is telling me the correlation is $-0.9939881.$ Look at the graph.