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!

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

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