1) How to find expectation of max of random variables , i.e : $\mathbb{E}[max(x_1,x_2,\dots,x_n)]$ where $x$ are IID random variables from $\mathcal{N}(\mu,\sigma^2)$.
- I know that CDF is $F(x)^n$ and PDF is $nF(x)^{n-1}f(x)$. I have also seen simplifications for uniform and exponential distributions but not for Gaussian distribution.
2) In general, How to solve
$\int_{x=0}^{\infty} [nx F(x)^{n-1}f(x)] dx$