Calculate expectation of Cumulative distribution function of a normal distribution

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?

1 Answer

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 $\frac12$.

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