How can I calculate this integral? $\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$

Question

Suppose we have the density and distribution of the standard normal. How can one calculate the integral:

$\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$

Note this is not included in the Wikipedia list of integrals of Gaussian functions.

Further, this is not the same as How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$ because that solution only works for a standard normal second term, i.e. $\phi (X)$ whereas this problem includes a coefficient and additional term. One cannot put into the same form as that case because $c$ and $e$ cannot be brought out of the normal density due to their involvement in the exponential component.

Answer 1

But if you know the form discussed in comments

$$\int_{-\infty}^{\infty}\Phi(a+bx) \phi( x ) dx=\Phi\left(\frac{a}{\sqrt{1+b^2}}\right)$$

it's quite straightforward!

Just do the substitution $Z = c+eX$ (don't forget the Jacobian), and use the fact there that you already know (but in $z$ rather than $x$).