Covariance$(X,Y) \geq 0$ if $X,Y \geq 0$?

Question

I was wondering if you can say something about the covariance of two positive variables $X$ and $Y$?

Answer 1

You cannot conclude that $\mathrm{Cov}(X,Y)\geq 0$ in general. To see this, let $X\sim\mathrm{bin}(1,p)$ and $Y=1-X$, then obviously $P(X\geq 0)=P(Y\geq 0)=1$ but $$ \mathrm{Cov}(X,Y)=\mathrm{Cov}(X,1-X)=-\mathrm{Cov}(X,X)=-\mathrm{Var}(X)<0 $$ if $p\in (0,1)$.

In general you have the bound (see e.g. this answer): $$ |\mathrm{Cov}(X,Y)|\leq \sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)} $$

Answer 2

Might be interesting and related to your question:

http://www.math.tu-dresden.de/sto/schmidt/dsvm/dsvm2003-4.pdf