Let $u,v\in\mathbb{R}^n$ and $a,b\in\mathbb{R}$. Let $\phi$ denote the standard Multivariate Gaussian PDF and $\Phi$ denote the standard Gaussian CDF. Does anybody know how to solve $$\int_{\mathbb{R}^n} \phi(u)\Phi\left(\frac{v^Tu - a}{b^2}\right)du$$
Any help is appreciated! Thanks ahead of time.
$$ \begin{align} \int_{\mathbb{R}^n} \phi(u) \Phi\left(\frac {v^T u - a} {b^2} \right)du &= \int_{\mathbb{R}^n} \phi(u) \Pr\left\{Z \leq \frac {v^T u - a} {b^2} \right\}du \\ &= \Pr\left\{Z \leq \frac {v^T U - a} {b^2} \right\} \\ &= \Pr\left\{b^2Z - v^TU \leq -a \right\} \\ &= \Phi\left( \frac {-a} {\sqrt{b^4 + v^Tv}}\right)\end{align} $$
where $Z \sim \mathcal{N}(0,1)$ is a standard univariate normal,
$U$ is the $n$-dimensional standard multivariate normal independent of $Z$
First equality use the definition of $\Phi$,
second one use the law of total probability,
third one just rearrangement,
and the last one is using the fact that independent multivarate normal random vector are jointly multivariate normal, and the affine transformation of multivariate normal is still multivariate normal, with univariate normal as a special case.