Is it possible to find the expected $\max_i(|x_i|)$, $\boldsymbol{x}\in \mathbb{R}^n$ where $x_i \overset{\text{iid}}{\sim}\mathcal{N}(\mu,\sigma^2)$, with known, finite mean and standard deviation? What about for $\mu=0$?
In general, I know that the min/max lies between $1/\sqrt{n}$ and $\sqrt{n}-1$ standard deviations from the mean, where $n$ is the sample size. Here, $n$ is the length of vector $x$ and I am trying to find the expected maximum magnitude of all elements in vector $\boldsymbol{x}$.