Estimating the distribution of the maximum of N values drawn from N different normal distributions Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with known mean $\mu_i$ and known standard deviation $\sigma_i$, $i=0,...,N-1$.
For each $i$, 1 random samples is drawn from from the distributions above $x_i \sim N(\mu_i,\sigma_i)$.
Let $X$ denote the set of these $x_i$ samples.
How does one estimate the distribution of $max(X)$? Does it require a Monte-Carlo simulation, or is there a better way (or even analytical way)?
 A: You do not need to estimate the distribution of $\text{max}(X)$, as one can find the distribution exactly. In particular, there is a very simple and elegant result that:

*

*the cdf  of the maximum is the product of the respective cdf's.

Then, the pdf of the max is obtained by simply differentiating the max cdf wrt $x$.
Automated version
While the above process is conceptually simple, it can get algebraically messy. An automated version exists as the Maximum function in the mathStatica add-on package for Mathematica. We are given parent variable $X \sim N(\mu, \sigma^2)$ with pdf $f(x)$:

To illustrate, let us suppose we have 3 variables $X_i \sim N(\mu_i, \sigma_{i}^2)$, with parameter values as shown:

Then, the pdf of $\text{max}(X_i)$ is:

[ The parameters could just as easily be symbolic -- and the pdf's do not need to share the same parent distribution. ]
The following diagram compares the theoretical pdf of the maximum just obtained (dashed red curve) with a Monte Carlo simulation of same  (squiggly blue line):

