# Probability Distribution of Infinity Norm of a Vector of Independent Random Variables

Assuming $V\in \mathbb{R}^k$ to be a vector of independent random variables each with $\sim\mathcal{N}(0,1)$ (Or even more general $\sim\mathcal{N}(a,b)$). I was wondering how I can calculate the distribution (so both $\mathbb{E}$ and $\sigma$) of infinity norm of my vector; i.e. $\|V\|_\infty=\max_{i\in \{1,2, \ldots, k\}} |V_i|$ ?

• Where you have ~$\mathcal N(0,1)$ I changed it to $\sim\mathcal N(0,1)$. That is standard. ${}\qquad{}$ – Michael Hardy Sep 29 '15 at 16:44
• Why repost this? Anyway the general case N(a,b) will be a mess but the "standard" case N(0,1) is rather readily, but partly, solved noting that, for every nonnegative $x$, $$P(\|V\|_\infty\leqslant x)=P(|V_1|\leqslant x)^k=(2\Phi(x)-1)^k,$$ and differentiating this to get the PDF. As a consequence, $$E(\|V\|_\infty)=\int_0^\infty(1-(2\Phi(x)-1)^k)dx.$$ – Did Sep 29 '15 at 16:44
• "Thanks. I reposted this because no one answered my previous question" This is explicitely discouraged (but perhaps you knowingly decided to go against the rules of the site simply for your convenience?). – Did Sep 29 '15 at 16:52
• @Did What if I have a bunch of vectors whose infinity norms are added to each other? I mean the case $P(\|V_1\|_\infty+\|V_2\|_\infty+\|V_3\|_\infty+...+\|V_N\|)_\infty<x)$? – Hossein K. Mousavi Sep 29 '15 at 16:55