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I have to calculate the expectation of the Cumulative Distribution Function of a normally distributed random variable X, which has variance 1 and mean 0. I calculated the integral of the CDF (taken as an RV distributed on [0,1] and X and got 0 as the answer. I am not sure if I've done this right. Is there a better way of solving this question?

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The expectation of the cumulative distribution function is independent of the distribution. The cumulative distribution function is uniformly distributed over $[0,1]$, so its expectation is $\frac12$.

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    $\begingroup$ For a continuously distributed variable this works. Not otherwise: for example a constant r.v. will give $1$ instead of $1/2$. $\endgroup$
    – Ian
    Apr 17, 2016 at 15:34

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