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That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}^Z\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1/\sqrt2\le 0,Y_1/\sqrt2\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$

That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}^Z\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1/\sqrt2\le 0,Y_1/\sqrt2\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$

That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}Ẑ\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1/\sqrt2\le 0,Y_1/\sqrt2\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$

That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}^Z\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1/\sqrt2\le 0,Y_1/\sqrt2\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$

That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}Ẑ\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$

That it relates to the standardised bivariate Normal orthant probability pointed out by wolfies is due to the fact that $$\begin{align*} \mathbb{E}^{X,Y}\left[ \Phi \left(X \right) \Phi \left(Y \right) \right] &= \mathbb{E}^{X,Y} \overbrace{\left[\mathbb{E}Ẑ\{\mathbb{I}(Z\le X)\}\mathbb{E}^W\{\mathbb{I}(W\le Y)\}\right]}^{Z,W\sim\mathcal{N}(0,1)}\\ &=\mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z\le X) \mathbb{I}(W\le Y)\right]\\ &= \mathbb{E}^{X,Y,Z,W} \left[\mathbb{I}(Z-X\le 0) \mathbb{I}(W-Y\le 0)\right]\\ &= \mathbb{E}^{X_1,Y_1} \left[\mathbb{I}(X_1\le 0) \mathbb{I}(Y_1\le 0)\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1\le 0,Y_1\le 0\right]\\&= \mathbb{P}^{X_1,Y_1} \left[X_1/\sqrt2\le 0,Y_1/\sqrt2\le 0\right]\\\end{align*} $$ where $(X_1,Y_1)$ is now a bivariate normal vector with correlation $\rho/2$: $$\mathbb{E} \left[X_1Y_1\right]=\mathbb{E} \left[(Z-X)(W-Y)\right]=\mathbb{E} \left[XY \right]=\rho$$ and $$\mathrm{var}(X_1)=\mathrm{var}(Y_1)=\mathrm{var}(Z)+\mathrm{var}(X)=2$$