Calculating $p(X_2-X_1+Y>\pi, X_2>\pi)$ with $X_1$ and $X_2$ correlated, jointly Normal and $Y$ independent from them and Normal

Question

Given $X_1$ and $X_2$ as correlated ($\rho$) and jointly Normal random variables (each distributed ~$N(0,\sigma_X^2)$), and also given $Y$ as Normally distributed((~$N(0,\sigma_Y^2)$)) and independent of $X_1$ and $X_2$, how can one compute

$p(X_2-X_1+Y>\pi, X_2>\pi)$

Any help and guidance would be greatly appreciated!

I know this is equivalent to

$p(X_2-X_1+Y>\pi, X_2>\pi)$=$p(X_2-X_1+Y>\pi | X_2>\pi)p(X_2>\pi)$

but I'm not sure if this is the way to proceed and how to do so. Perhaps there is a simple way to do this that I'm missing? I would really appreciate the help!

Answer 1

This is $$ E\left(\Phi\left(\frac{X_2-X_1-\pi}{\sigma_Y}\right)\cdot\mathbf 1_{X_2\gt\pi}\right). $$ Now, use the density of $(X_1,X_2)$ to transform this expectation into a double integral on $\mathbb R\times(\pi,+\infty)$.