Covariance between a ratio and its denominator

Question

Suppose we have two positively correlated and positive random variables $X$ and $Y$.

That is, $X>0$, $Y>0$ surely and $cov(X,Y)>0$.

Questions: If $cov(X/Y,Y)=0$, does it imply $X/Y=c$ (a constant)?

p.s. I have asked a specific question but I suppose a general (and useful) expression of the covariance will be good too.

Answer 1

No.

Suppose the following joint events each have probability $\frac14$:

• $X=1,Y=1, X/Y=1$
• $X=2,Y=1, X/Y=2$
• $X=2,Y=2, X/Y=1$
• $X=4,Y=2, X/Y=2$

then $X>0$, $Y>0$ surely and $cov(X,Y)=\frac38>0$, but $cov(X/Y,Y)=0$