All Questions
Tagged with pr.probability geometric-probability
29 questions
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,...
- 309
12
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 $k < n$ ...
- 420
5
votes
3
answers
5k
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 ...
- 6,853
47
votes
7
answers
5k
views
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 ...
- 11.4k
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 ...
- 335
11
votes
1
answer
746
views
Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)
Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$.
Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$.
Find ...
- 3,557
28
votes
5
answers
2k
views
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 ...
- 19.7k
14
votes
3
answers
4k
views
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 $p=2$:...
- 667
13
votes
1
answer
762
views
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
- 5,470
10
votes
4
answers
904
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 ...
- 121
10
votes
3
answers
2k
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 ...
- 193
10
votes
3
answers
5k
views
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 ...
- 371
10
votes
2
answers
797
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 ...
- 1,125
10
votes
2
answers
847
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{...
- 1,127
7
votes
1
answer
310
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 ...
- 169
7
votes
1
answer
757
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 ...
- 367
6
votes
3
answers
447
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) = ...
- 6,853
6
votes
1
answer
264
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 ...
6
votes
2
answers
497
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 ...
- 826
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( \...
- 519
4
votes
1
answer
1k
views
Defining a probability distribution on each tangent space of a manifold?
I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to ...
- 518
4
votes
1
answer
447
views
Area enclosed by Brownian motion (without winding number)
The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
- 29.8k
4
votes
1
answer
116
views
Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$
Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and ...
- 4,049
3
votes
1
answer
182
views
How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
3
votes
2
answers
590
views
The Largest Piece of Circumference
We add $n$ random lines to the unit disc. We do so by adding two points on the disc, transcribing a line between them and extending that line to the boundary of the disc, namely the unit circle. Each ...
- 133
2
votes
2
answers
322
views
Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
- 6,853
0
votes
2
answers
534
views
Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
- 6,853
0
votes
1
answer
292
views
Volume of randomly changing sphere follows beta distribution
We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
- 329
0
votes
0
answers
113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
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