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Find $Cor(\Phi(X),\Phi(Y)),$ where $X,Y$ are standard normal with $\operatorname{Cor}(X,Y)=\rho$, $ \Phi(x)=\int_{-\infty}^x\phi(x)dx$ and $\phi(x)$ is the standard normal density.

Using the substitution $x=\rho y+\sqrt{1-\rho^2}t$, I was able to simplify it a bit and now need to integrate the following $$E[\Phi(X)\Phi(Y)]=\frac 1{2\pi}\int_{\mathbb R}\int_{\mathbb R}\Phi(\rho y+\sqrt{1-\rho^2}t)\Phi(y)\phi(t)\phi(y)\,dt\,dy$$

Any hints how to proceed? I know $\Phi(X)$ follows uniform distribution

PS. The final answer is $\frac 6\pi\arcsin\frac\rho2$

Bernard
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Anvit
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    $(X,Y)$ is supposed to be jointly normal for this result to hold. See https://math.stackexchange.com/questions/2954999/finding-correlation-between-cdf-of-two-normal-distributions?noredirect=1&lq=1,https://math.stackexchange.com/questions/2550103/find-the-correlation-coefficient-rho-g-of-gx-hy?noredirect=1&lq=1. – StubbornAtom Apr 29 '19 at 08:35
  • @StubbornAtom What does it mean to say $X,Y$ are jointly normal? – Anvit Apr 29 '19 at 08:36
  • It means $(X,Y)$ has a multivariate normal distribution ( 2D case). This is not the same as saying $X,Y$ are standard normal with correlation $\rho$. It looks like you have assumed $X,Y$ to be independent in your attempt which is certainly not the case here (unless $\rho=0$). – StubbornAtom Apr 29 '19 at 08:40

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