What is the distribution of the random variable $\max(x)$?

Question

$x$ is a random vector that follows the $p$-dimensional normal distribution $\mathcal{N}(0,I_d)$. What is the distribution of the random variable $\max(x)$ ?

Answer 1

Hints:

1. $\ \max_\limits{1\le i\le p}x_i \le y\ $ if and only if $\ x_i\le y\ $ for all $\ i=1,2,\dots, p\ $.
2. $\ x_1, x_2, \dots, x_p\ $ are independent.