Suppose $X \sim N(0, I)$. How would I go about calculating \begin{align*} \mathbb{E}[\Phi(\Delta_1 + \theta_1^\intercal X)\Phi(\Delta_2 + \theta_2^\intercal X)]? \end{align*} where $\Delta_1, \Delta_2 \in \mathbb{R}, \theta_1, \theta_2 \in \mathbb{R}^p$. Here, $\Phi(x) = (2\pi)^{-1/2}\int_{-\infty}^{x}\exp(-t^2/2)dt$ is the standard normal CDF. I've found previous posts for which only one $\Phi$ was present, but not both, and the strategy from one doesn't follow with two now.


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