Let X and Y be 2 random variables with $\displaystyle |\mathbb{E}[X]|, |\mathbb{E}[Y]|, $and $\displaystyle |\mathbb{E}[\frac{X}{Y}]| $ all finite, and with $\displaystyle \mathbb{P}(Y = 0) = 0 $ and $\displaystyle \mathbb{E}[Y] \ne 0 $.

Prove that $\displaystyle \mathbb{E}[\frac{X}{Y}] = \frac{\mathbb{E}[X]}{\mathbb{E}[Y]}$ if and only if Cov$\displaystyle (Y,\frac{X}{Y}) = 0 $