I am trying to figure out if there is a closed form expression for the following expectation:
$\int\int \phi(\gamma_1)\phi(\gamma_2) \mathcal{N}(\gamma\big|\mu, \Sigma)d\gamma_1 d\gamma_2$
where $\gamma = \left[\gamma_1, \gamma_2\right]^T$ and $\phi(\cdot)$ is the standard normal CDF. I am interested in the general case with an arbitrary covariance matrix.
I know the following result holds:
$\int \phi(\gamma_1) \mathcal{N}(\gamma\big|\mu_1, \Sigma_{11})d\gamma_1 = \phi\left(\frac{\mu_1}{\sqrt{1+\Sigma_{11}}}\right)$
but is there a way to extend this to the 2 dimensional case?
I will appreciate any help or suggestions.