What is the correlation of a copula of a multivariate normal distribution?

Question

If $X_1, X_2 \sim \mathcal{N}(0,1)$ with correlation $\rho$, and $Y_1 = F_{X_1}(X_1)$ and $Y_2 = F_{X_2}(X_2)$, what is the correlation between $Y_1$ and $Y_2$? What is a proof that the correlation of $X_1, X_2$ is tightly coupled with the correlation of $Y_1, Y_2$, i.e. when the correlation in $X$ is low/high, so will be the correlation in $Y$?

Answer 1

Edit: I could have saved time by giving the following link: Finding correlation between CDF of two normal distributions

The easy part is to calculate the mean and variance of the $Y_i$'s, which is actually straightforward from the fact that for any random variable $X$ with CDF $F_X(\cdot),$ it holds that $F_X(X)\sim U(0, 1).$ Hence $E[Y_1] = 1/2 = E[Y_2]$ and $Var[Y_1] = 1/12 = Var[Y_2].$

Also note that since marginal distributions of $X_i$'s is the standard normal, their CDF's $F_{X_i}$'s are same as the standard normal CDF, commonly denoted as $\Phi(\cdot).$

The slightly difficult part is to calculate $E[\Phi(X)\Phi(Y)].$ This is in fact a standard result, and the answer is given by $$E[\Phi(X) \Phi(Y)] = \frac{1}{4} +\frac{1}{2\pi} \sin^{-1} \frac{\rho}{2}.$$

Using the above results we can easily get the desired correlation between $Y_1$ and $Y_2.$