Let $a$ be a constant, X is a standard normal and $\Phi$ is the c.d.f. of standard normal.
- what is the density of $Y$ where $$Y = \Phi(aX)$$
- what is the density of $$Y=\Phi(X + a)$$
The second case seems to me that doesn't have a clean density, if so, we can skip this part as it will not be too helpful. For the first case, defining as $\phi(x)$ the density of a standard normal, in my approach i get to $$\phi\Large(\small\frac{\Phi^{-1}(x)}{a}\Large)\small \frac{1}{a\phi(\Phi^{-1}(x))} $$ but then i'm not sure how to proceed. Considering $\Phi^{-1}(x) = y$ as is the same function inside the two density i get $$\frac{1}{a}\large e^{\small-\frac{y^2}{2\frac{a^2}{1-a^2}}}$$but this form doesn't look too good. The final purpose is to then compute $$E[\Phi(aX)]$$ which i know to be equal to $\frac{1}{2}$. See Expectation of normal c.d.f. of a normal random variable