Let $Z$ be a standard normal random variable and denote by $\Phi$ its cumulative distribution function. Let $a$ be a real number. Question: what is the expected value of the random variable $\Phi(Z+a)$?
Now, if $a=0$, the answer is simple. Since $\Phi$ is continuous, a theorem states that $\Phi(Z)$ has the standard uniform distribution, hence its expected value is $1/2$.
But what happens when $a\neq 0$? I tried to approach the problem transforming the random variable $\Phi(Z+a)$ in a similar way, possibly with some translation, but I got nowhere. The issue here is that also the random variable $\Phi(Z+a)$ takes its values in the interval $[0,1]$ but it cannot be anymore a uniform random variable, so I find problematic to approach this problem in the same way as in the case $a=0$.
Thank you in advance for your ideas.