2
votes
0answers
32 views

Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ ...
0
votes
0answers
36 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
2
votes
1answer
426 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
2
votes
1answer
91 views

Expected value of maximum distance between points

Consider a two dimensional square domain ($S$) of size $l \times l$. We generate a point $\mathbf{x}_i = (x_i,y_i)$ in S with uniform distribution, i.e., the point is equally likely to be anywhere ...
1
vote
0answers
153 views

References for problems related to uniformly distributed points/ arcs on circle

I am looking for references to problems related to computing the statistical properties of uniformly randomly chosen points or arcs on the unit circle, e.g.: On a unit circle, $n$ points are ...
1
vote
0answers
68 views

Probability of a certain circular configuration

Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - ...
-1
votes
1answer
86 views

Probability that points are on a straight line

I am looking at a formula to calculate the probability that $n$ points are on a straight line between point $1$ and point $n$ in 2d Euclidean space. If the points are exactly on the line, the ...
87
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
3
votes
2answers
1k 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 ...
1
vote
1answer
101 views

A question about a metric space

I have a question regarding the following problem. Let $A,B,C$ be the three independently selected, uniformly distributed points on the unit sphere $S^3$ in $\Bbb R^4$. What's the probability ...
13
votes
3answers
373 views

What is the expected area of a polygon whose vertices lie on a circle?

I came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly ...
1
vote
2answers
192 views

Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
2
votes
1answer
149 views

Probabilities of Non-Regular Dice

Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die. If I have an arbitrary 6-sided solid, how do you ...
3
votes
0answers
89 views

Computing the proportion of vectors with the same sign

Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...