Start with a random variable $r \sim \mathcal{N}(0,1)$.
Now consider the random variable $\sigma(r)$ formed by passing it through a standard logistic function $\sigma(x) = \frac{1}{1 + e^{-x}}$. I would like to know the mean and variance of $\sigma(r)$.
The mean value is 0.5 by symmetry, but the variance is trickier. By definition $$\mathrm{Var}[\sigma(r)] = \mathbb{E}[\sigma(r)^2] - \mathbb{E}[\sigma(r)]^2$$ and $\mathbb{E}[\sigma(r)]^2 = 0.25$.
Is there a way to calculate $\mathbb{E}[\sigma(r)^2]$ analytically?