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suppose X has normal distribution with $\mu=\sigma^2=1$. how can calculate $E(\Phi(X-1))$

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Because the question is self-study, I will only provide hints.

If $X$ has Normal distribution with mean = 1 and variance = 1, then what distribution does $X-1$, call it $Y$, have?

Can you see how to apply the techniques used in inverse transform sampling https://en.wikipedia.org/wiki/Inverse_transform_sampling in order to determine the probability distribution of $\Phi(Y)$, where of course $\Phi$ is the standard Normal cumulative distribution function? Once you have done that, computing the expected value of a random variable having that distribution should be easy. You can of course use simulation to check your answer.

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  • $\begingroup$ Yes that makes it too easy. $\endgroup$ – Michael R. Chernick Mar 5 '17 at 20:55

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