I'm stuck with this problem: Calculate $\mathbb{E}[N(X)]$, where N(·) is the cdf of the standard normal distribution, and X is a standard normal random variable.
Here's where I'm stuck: We know that:
$$\mathbb{E}[g(X)]=\int_{-\infty }^{\infty}g(x)f_{X}(x)dx$$
And we are given that:
$$ g(x) = N(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x}e^{-\frac{u^2}{2}}du $$
$$ f_{X}(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} $$
So, the solution should be to develop this:
$$\begin{split} \mathbb{E}[N(X)] &= \int_{-\infty }^{\infty} \left( \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x}e^{-\frac{u^2}{2}}du \right) \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \right) dx\\ &= \frac{1}{2 \pi} \int_{-\infty }^{\infty} \left( \int_{-\infty}^{x}e^{-\frac{u^2}{2}}du \right) e^{-\frac{x^2}{2}} dx\\ &= \frac{1}{2 \pi} \int_{-\infty }^{\infty} \int_{-\infty}^{x} e^{-\frac{u^2+x^2}{2}}dudx \end{split}$$
My questions are:
- Can we combine the exponents moving the term with x inside, the last step above?
- If yes, this looks like a good opportunity to change variables to polar. If that's the case, how to change the limits of the integrals from $x$ and $u$ to $r$ and $\theta$?
Any help would be extremely appreciated! Thanks!