Conditional expectation of normal cdf Suppose $X \sim N(0, \sigma^2)$, is it possible to evaluate $E[\Phi(\frac{aX + b}{c}) | X > k]$ in closed form, where $\Phi$ is the standard normal cdf?
The motivation comes from that it is possible to evaluate something like $E[\Phi(\frac{aX + b}{c})]$ with a closed-form expression. But not sure if something similar holds for the conditional expectation case.

$$E\left[\Phi\left(\frac{aX + b}{c}\right) | X > k\right] = \left[1-\Phi(k/\sigma)\right]^{-1}\int_{k}^{\infty}\frac 1 {\sigma}\phi\left(x/\sigma\right)\Phi\left(\frac{ax + b}{c}\right)dx$$
This can be solved, check https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions
You will have of course to make transformations, change of variables, etc.
Look up also Owen 1980 cited in the wikipedia article, it contains more indefinite integrals, which is what you need.
