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### Average minimum distance between $n$ points generate i.i.d. uniformly in the ball

Let $U \in \mathbb{R}^3$ be distributed uniformly in the Ball in $\mathbb{R}^3$ centered at zero. That is $U \sim f_U(u)= \frac{1}{ \frac{4}{3} \pi R^3}$ for all $\|u\|\le R$ where $R$ is the ...
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### Mean distance between matrix entries

Given a $4$ by $4$ matrix, or in general an $n$ by $n$ square matrix, can we determine the mean euclidean distance (i.e. $\sqrt{\Delta x ^2 + \Delta y^2}$) between entries that are not neighbours? ...
### Evaluate $\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx$
This page contains an interesting identity$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$ ...
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,....