Let $X,Y$ are two random variables which are not necessarily independent. It is easy to get $\mathbb{E}(X)$ ann $\mathbb{E}(Y)$. I want to know: is there some approximation to $\mathbb{E}(\frac{X}{Y})$?
[Update] The background is that I want to calculate the expectation of Pearson product-moment correlation coefficient - $\mathbb{E}(\rho_{xy})$.
The Pearson product-moment correlation coefficient is $\rho_{xy} = \frac{Cov_{xy}}{\sigma_x\sigma_y}$
It is easily to get $\mathbb{E}(Cov_{xy})$ and $\mathbb{E}(\sigma_x\sigma_y)$, so I want to know a approximation to get $\mathbb{E}(\rho_{xy})$ from the above two value.