Linked Questions

6 votes
1 answer
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&...
Demetri Pananos's user avatar
3 votes
1 answer
403 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 ...
Will Sherwood's user avatar
125 votes
27 answers
13k views

Unexpected examples of natural logarithm

Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common ...
José Carlos Santos's user avatar
64 votes
11 answers
70k 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?
Kenshin's user avatar
  • 2,082
26 votes
2 answers
33k 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 ...
Alecto Irene Perez's user avatar
11 votes
1 answer
11k 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 ...
CAFxX's user avatar
  • 427
18 votes
1 answer
4k 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{\...
Martin Thoma's user avatar
  • 9,569
10 votes
1 answer
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 ...
Boby's user avatar
  • 5,779
6 votes
2 answers
1k 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? ...
user929304's user avatar
  • 1,414
10 votes
1 answer
589 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$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$ ...
Infiniticism's user avatar
  • 8,503
0 votes
0 answers
158 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,....
Martin Thoma's user avatar
  • 9,569
1 vote
0 answers
73 views

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....
MarcosMFlores's user avatar
1 vote
0 answers
56 views

Average distance between points

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 $\frac{x}{\sqrt{y}}$, ...
someguy's user avatar
  • 13
0 votes
0 answers
35 views

Why is the average entry of this matrix lower than expected?

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}...
ty.'s user avatar
  • 540