# $\mathbb{E}[\sigma(r)^2]$ with $r \sim \mathcal{N}(0,1)$

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?

• With is the computation of the mean so obvious? – Davide Giraudo Jan 30 '19 at 11:40
• I'm pretty sure it follows from the symmetry of $r$ and $\sigma$. – prdnr Jan 30 '19 at 11:41
• I see. The key point is that $\sigma(x)+\sigma(-x)=1$, but I do not know whether a similar relation holds for $\sigma^2$. – Davide Giraudo Jan 30 '19 at 14:58
• Yes, $\sigma^2$ is not symmetric so there I doubt there's any similar identity – prdnr Jan 30 '19 at 15:16
• According to the wiki page for the logit-normal (the distribution you're describing), there are no analytical formula for the moments en.wikipedia.org/wiki/Logit-normal_distribution. However, you could easily get a monte carlo approximation with a few lines of R. – aleshing Jan 30 '19 at 15:57