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2k views

### 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? ...
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### 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$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$ ...
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### What is the expected maximum distance of a set of two randomly sampled points in [0, 1]^n?

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,....
1 vote
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### Average of the inverse distance between two points in a non-uniform disc.

Suppose I have two position vectors $\mathbf{r}_1$ and $\mathbf{r}_2$. The magnitude of both of these vectors are chosen independently from the same probability distribution of known mean and variance....
1 vote
If there were a square field that is $x$ km wide and long, and there are $y$ objects in the field, is there a way to calculate the average distance between each object? I tried $\frac{x}{\sqrt{y}}$, ...
Suppose I have a family $X$ of vectors $X_i \in \mathbb{R}^2$, where $i\in \{1,\dots, N\}$. Let $U(a,b)$ be the continuous uniform distribution with bounds $a,b.$ They are such that \$X_i=(X_{i0},X_{i1}...