Continuing this question:
I sample two points from the $n$-dimensional unit cube: $$p_{i,1}, p_{i,2} \sim U([0, 1]^n)$$
Now I do this $N$ times. I define the maximum distance as
$$m_d := \max_{i=1,..., N}(|p_{i,1} - p_{i,2}|_2)$$
What is the expected maximum distance $\mathbb{E}(m_d)$ for $N$ pairs of 2 points?