# Expected value of a non-linear function of a normal random variable

Consider a random variable $$X\sim N\left(\mu,\sigma^2\right)$$, and a monotonically increasing non-linear function of $$X$$, call it $$Y=f\left(X\right)$$, defined as: $$Y=f\left(X\right)=\Phi\Big(a-b\,\Phi^{-1}\left(X\right)\Big)$$ where $$a$$ and $$b$$ are real, non-negative numbers, $$\Phi\left(\cdot\right)$$ indicates the standard normal CDF, and $$\Phi^{-1}\left(\cdot\right)$$ indicates its inverse. Is it possible to express the expected value $$\mathbb{E}\left[Y\right]$$, i.e.: $$\mathbb{E}\Bigg[\Phi\Big(a-b\,\Phi^{-1}\left(X\right)\Big)\Bigg]$$in closed form?

I'm aware of a formula for the expected value of $$\Phi\left(\cdot\right)$$ when its argument is a linear function of a normally-distributed r.v. (e.g. Expected Value of Normal CDF), but I really don't know how to cope with this particular non-linear case. For the choice of paramenters $$a=0$$ and $$b=1$$ the solution should be straight-forward, so maybe there exists a way to deal with this more general problem. Any ideas and/or suggestions? Many thanks in advance!

• What is $\Phi^{-1}(X)$ for values of $X$ outside of $[0,1]$? – kimchi lover Apr 1 at 23:26
• @kimchilover: Oops, right. – Nate Eldredge Apr 1 at 23:28