${\rm Cov}(X,Y)=1/4 \times ({\rm Var}(XY)-{\rm Var}(X/Y))$

Question

I saw that formula in this paper, equation (1.3).

The formula should hold for all the random variables such that those functionals are properly defined. I have been trying to prove it, but without success.

Answer 1

It seems to me that that identity is not true in general. Consider for instance $ X = Y$. Then we'd have

$$ \text{Var}(X) = \frac{1}{4} \text{Var}(X^2) $$

but if, e.g. $\mathbb{P}(X=1) = \mathbb{P}(X=2) = 1/2$, that is false.