Suppose $F$ is the cumulative distribution function of the normal distribution with mean $a$ and standard deviation $b$, and suppose $f$ is the probability density function of the normal distribution with mean $w$ and standard deviation $z$. How can I calculate this integral in closed form?
$$\int F(x \mid a,b)f(x \mid w,z) {}dx$$
It seems like my question is very similar to this one, but crucially I'm not using that standard versions of the cdf or pdf. I tried converting my functions to their standard forms, but ended up with this:
$$\int \Phi\left(\frac{x-a}{b}\right)\frac{\phi\left(\frac{x-w}{z}\right)}{z} {}dx$$
And now I'm really confused!
P.S.: If the above does have a closed-form simplification, my true problem is actually closer to this:
$$\int F(2a - \xi - x \mid a,b)f(x \mid w,z) {}dx$$