All Questions
Tagged with plane-geometry pr.probability
10 questions
24
votes
3
answers
4k
views
What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?
Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay ...
- 4,922
12
votes
1
answer
400
views
Probability that random cubic polynomials meet in a square
Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...
- 151k
9
votes
1
answer
338
views
Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
- 151k
6
votes
1
answer
273
views
Proof of a statement from Steele's "Probability theory and combinatorial optimization"
I am reading "Probability theory and combinatorial optimization" by J.M. Steele and am hung up on a statement made in Section 2.2 of Chapter 2, "Easy size bounds", in which it is stated (paraphrasing ...
5
votes
2
answers
323
views
Distribution over Penrose Tilings?
The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow ...
- 3,979
3
votes
1
answer
128
views
Random quads visible from a random point
Although the MO question Limit of lights in rooms was quickly closed,
it suggests a related question:
Q0. What is the probability that a random quadrilateral $Q$
is entirely illuminated from a ...
- 151k
3
votes
0
answers
144
views
What is the probability that these four random areas can yield a tetrahedron?
This is inspired by this problem about randomly broken sticks that can form a triangle. It goes in a different direction than this generalization about randomly broken sticks that can form a ...
- 13.4k
2
votes
1
answer
305
views
Distribution of area of randomly placed circles
I've searched the web now for ages to try and find a paper on the asymptotic distribution of the area of the union of randomly placed discs on the plane. Ideally, I would be looking for the discs to ...
- 903
2
votes
1
answer
121
views
Expected length of a certain kind of nearest-neighbor graph
Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...
0
votes
1
answer
176
views
Expected area of a pentagon formed from a randomly broken stick [closed]
Suppose we break a stick of length one at four randomly and independently chosen points and that the resulting pieces form a pentagon.
Such a pentagon can be formed with probability $1-(5/16) = {11\...
- 133