Questions tagged [geometric-probability]
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213
questions
47
votes
7
answers
5k
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Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
30
votes
8
answers
3k
views
A variation of the law of large numbers for random points in a square
I uniformly mark $n^2$ points in $[0,1]^2$. Then I want to draw $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point. Surely, for a given ...
30
votes
2
answers
1k
views
Shortest path through $\sqrt{n}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
30
votes
1
answer
1k
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Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
27
votes
5
answers
2k
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Moments of area of random triangle inscribed in a circle
The $2m$th moment of the (random) area of the triangle whose vertices are three independent, uniformly distributed random points on the unit circle appears to be $((3m)!/(m!)^3)/16^m$. Can anyone ...
25
votes
1
answer
1k
views
How random are unit lattices in number fields?
I was wondering how random unit lattices in number fields are. To make this more precise:
If $K$ is a number field with embeddings $\sigma_1, \dots, \sigma_n, \overline{\sigma_{r+1}}, \dots, \...
15
votes
2
answers
743
views
Random noncrossing chords of a circle
Suppose you generate random chords of a circle, with endpoints selected uniformly over the circumference, rejecting any chord that crosses a previously generated chord.
The disk is then partitioned ...
14
votes
3
answers
4k
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How to generate random points in $\ell_p$ balls?
How do I feasibly generate a random sample from an $n$-dimensional $\ell_p$ ball? Specifically, I'm interested in $p=1$ and large $n$. I'm looking for descriptions analogous to the statement for $p=2$:...
14
votes
2
answers
317
views
Shortest path through $n^{1/3}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
14
votes
0
answers
367
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
13
votes
1
answer
946
views
A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
13
votes
4
answers
528
views
Alignment of random points
Whenever I draw randomly about ten points, I see that there will be always 3 points that are "almost" collinear. This observation leads me to considering the following questions:
Question 1: Suppose $...
12
votes
2
answers
898
views
The metric of the expected difference of random variables
Suppose we have a set of independent random variables $X_1,\ldots,X_n$ over $\mathbb{R}$.
It is easy to see that
$$d_{ij}=E[|X_i-X_j|]$$
satisfy the triangle inequality.
Is there any study of such ...
11
votes
6
answers
3k
views
Marginal density of uniform spherical distribution
Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ ...
11
votes
2
answers
955
views
Clique sizes in a unit disk graph
This is a spiritual successor to a question that Peter Shor answered here:
Generalized Euclidean TSP
Are there any results known on the asymptotic behavior of clique sizes in a unit disk graph with ...
11
votes
1
answer
304
views
Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube.
Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly random locations, i.e.,...
10
votes
3
answers
1k
views
Mean maximum distance for N random points on a unit square
Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...
10
votes
4
answers
897
views
The distribution of the shortest path through $n$ points
In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...
10
votes
1
answer
537
views
Can one use Brownian motion to prove that two manifolds are not conformally equivalent?
Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a ...
10
votes
2
answers
790
views
Fitting a mesh to a density function
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
10
votes
2
answers
803
views
Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
10
votes
3
answers
5k
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Mean minimum distance for K random points on a N-dimensional (hyper-)cube
Given K points in a N-dimensional (hyper-)cube with all edges length 1.
What is the expected minimal distance between 2 points.
I found the 1-dimensional case in this topic: Mean minimum distance for ...
10
votes
1
answer
466
views
Stochastic Covering Number of a Convex Set
Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ ...
9
votes
2
answers
422
views
Density of a saturated random packing of congruent circles
The problem of the expected density of a saturated random packing of unit circles in the plane can be described as follows.
In a circular region $C$ of a large radius pick a point at random and draw ...
8
votes
3
answers
1k
views
Taking points uniformly inside a general finite geometric domain
It is well known that if we want to take $n$ uniformly and randomly points inside a circle of radius $r$ and centered at the origin the following apparently correct approach for generating $x$ and $...
8
votes
2
answers
253
views
What is the probability that these sets intersect?
Let $A$ be the subset of $\mathbb{R}^n$ defined by $A=\{x\in\mathbb{R}^{n}:|x_{1}-x_{n}|+\sum_{i=1}^{n-1}|x_{i+1}-x_{i}|\leq d\}$ for a given $d$. Next, sample a point $p$ uniformly in the unit cube, ...
8
votes
2
answers
938
views
Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$
Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid
$$\operatorname{KL}(p_\theta\...
8
votes
3
answers
1k
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Introduction to information geometry and/or geometric control theory
Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
7
votes
3
answers
2k
views
Packing density of randomly deposited circles on a plane
Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$:
Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...
7
votes
1
answer
753
views
Length of nearest neighbor path in travel salesman problem
Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...
7
votes
1
answer
303
views
Iterating projections to random halfspaces
Consider the following process:
Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \...
7
votes
0
answers
137
views
Probability of landing inside the convex hull of previously sampled points
Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$.
I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
6
votes
3
answers
703
views
Expected absolute value of the average of two points from the disc
Looking at Average distance of the mean of n random complex numbers in a unit disc, I tried to figure out
what is the expected absolute value $|\frac{z_1 + z_2}{2}|$ of two numbers $z_1, z_2\in\...
6
votes
3
answers
1k
views
The distribution of the number of chord intersections
This is a follow-up to this question: Given $n$ random chords of a circle, what is the distribution of the number of intersections? Random is defined by "endpoints uniform on the circle".
Update ...
6
votes
2
answers
1k
views
Definition of random measures
Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from?
I would like to have a motivation ...
6
votes
1
answer
275
views
Local Lipschitzness of parameterization of Gaussians in Wasserstein space
Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and ...
6
votes
2
answers
480
views
Average distance of the mean of $n$ random complex numbers in a unit disc
Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
6
votes
1
answer
317
views
Almost evenly distributed spherical random vectors
Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...
6
votes
3
answers
293
views
Probability of random geodesics on the half-sphere intersecting
4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...
6
votes
3
answers
415
views
Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace
Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
6
votes
1
answer
180
views
Does there exist a Penalized Conditional Expectation?
In my recent work I've become interested in working with the minimizer of
$$
\mathbb{E}[(Y-Z)^2] + \lambda P(Z),
$$
$Y$ is an observed random variable, $P$ is a positive-convex penalty function, $Z$ ...
6
votes
1
answer
232
views
Which orthant probabilities are the largest? (For a multivariate normal distribution)
I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
5
votes
3
answers
4k
views
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
5
votes
1
answer
460
views
The expectation of two sides of rectangle is equal. Can we deduce that in the expectation the rectangle is not very far from being a square?
Let $T$ be a set of $n\ge 3$ points in the plane such that not all of them lie in a common line. Pick two distinct points $\{a=\left( \begin{array}{c} a_{1} \\a_{2} \end{array} \right) ,b=\left( \...
5
votes
4
answers
855
views
Probability that convex hull of multivariate Gaussian sample contains a given point
I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls ...
5
votes
1
answer
300
views
What is the probability that a random chord in a sphere touches opposite hemispheres?
(edited) Consider the unit sphere $\mathbb{S}^2\subset \mathbb{R}^3$, and its upper $(z>0)$ and lower $(z<0)$ hemispheres.
Draw two independent, uniformly distributed points $X,Y$ on $\mathbb{S}^...
5
votes
1
answer
187
views
Probability of gaps between coordinates of a random point on the sphere
Let $X=(X_1,\ldots,X_n)$ be a point chosen uniformly at random from the sphere $S^{n-1}\subseteq \mathbb R^n$. Given $a>0$, what is the probability that $|X_1|^2-|X_i|^2\geq a$ for all $i>1$? Is ...
5
votes
1
answer
544
views
Show that the Markov chain of random tiling is irreducible
Consider a Markov chain on a state-space which is slightly weird: It is the space of all tilings of a hexagon as shown in the left-hand side of the figure below with three types of rhombi: yellow, ...
5
votes
1
answer
828
views
Hausdorff distance is a lower (or upper bound) for what probability metric?
In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|_A, \mu|_B),
$$
where
$d(A, B):= \...
5
votes
1
answer
365
views
Sums of uniformly random vectors from the $n$-dimensional unit ball
I'm interested in some instances of the following problem.
Let $n \geq 2$, and suppose we draw $k \geq 2$ vectors $v_1, \dots, v_k$ uniformly at random from the $n$-dimensional ball of radius $1$, $...