Linked Questions

6
votes
1answer
2k views

Average distance between two points on a unit square. [duplicate]

Consider the unit square $S =[0,1]\times[0,1]$. I'm interested in the average distance between random points in the square. Let $ \mathbf{a} = \left< x_1,y_1 \right>$ and $ \mathbf{b} = \left&...
3
votes
1answer
296 views

What is the average distance of two points chosen uniformly on a unit square? [duplicate]

What is the average distance of two points chosen uniformly on a unit square? What I am asking is how to calculate $E\left(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\right)$ for $x_1, x_2, y_1, y_2$ spread ...
49
votes
11answers
47k views

Average Distance Between Random Points on a Line Segment

Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?
21
votes
2answers
26k views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
10
votes
1answer
9k views

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
16
votes
1answer
3k views

How is the distance of two random points in a unit hypercube distributed?

Let $p_1, p_2 \sim U([0, 1]^n)$ with $n \in \mathbb{N}$ be two points in the $n$-dimensional unit hypercube which are uniform randomly independently sampled. How is the distance $d(p_1, p_2) = \sqrt{\...
8
votes
1answer
1k 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 ...
6
votes
2answers
523 views

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? ...
10
votes
1answer
322 views

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$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$ ...
0
votes
0answers
79 views

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,....