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Is there a closed form solution to $\int_0^1 \Phi^{-1}(u)\cdot \phi(\frac{u-y}{\sigma})du$, where $\phi$ is the pdf of a standard normal distribution and $\Phi^{-1}$ the inverse of its cdf?

For context: I am trying to find the posterior mean for a prior $x\sim\mathcal{N}(\mu,\omega^2)$ and likelihood $y\sim\mathcal{N}(\Phi(x),\sigma^2)$.

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    $\begingroup$ Given that $\Phi^{-1}$ is undefined for $u<0$ and $u>1$, this doesn't make much sense... $\endgroup$ – Tim Mar 10 '18 at 23:08
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    $\begingroup$ While I have no idea what this is for, I'm going to go out on a limb and suggest that this may not be the equation you're after and that you may have made a mistake getting to this point. $\endgroup$ – Ingolifs Mar 11 '18 at 0:40
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    $\begingroup$ Thank you for your feedback, I indeed scaled u to be in $[0,1]$, so the limits should be accordingly. I changed the question to reflect this. $\endgroup$ – user449277 Mar 11 '18 at 20:28
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    $\begingroup$ I have reopened the question, because now we do know what you're asking, but some explanation of the origin or intended interpretation of this integral might help you, due to the possibility that some mistake might have been made. $\endgroup$ – whuber Mar 11 '18 at 22:53
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    $\begingroup$ Thank you for the feedback, I have amended the question with the desired context! $\endgroup$ – user449277 Mar 16 '18 at 16:10

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