26
votes
Accepted
A variation of the law of large numbers for random points in a square
Given $n^2$ i.i.d. uniform points in $[0,1]^2$, the goal is to draw a configuration of $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point....
- 14.2k
18
votes
Moments of area of random triangle inscribed in a circle
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- 128k
16
votes
Accepted
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)
The conjecture is true, and it can be verified with fedja's method developed for your earlier question. We present a simplified version of the method. The idea is to scale the triangle so that one of ...
- 105k
15
votes
Moments of area of random triangle inscribed in a circle
The result follows from Dyson's Conjecture (A) from Part I, p.151 of Dyson, Freeman J., Statistical theory of the energy levels of complex systems. I-III, J. Math. Phys. 3, 140-156, 157-165, 166-175 (...
- 13.3k
15
votes
Moments of area of random triangle inscribed in a circle
If $q$ is uniformly distributed on the unit circle, $E[q^n]=\delta_{n0}$. So if $f$ is meromorphic, $E[f(q)]$ is the constant term in $f$.
The area of triangle $pqr$ is $\frac12\Im[(q-p)(\bar{r}-\bar{...
- 8,271
14
votes
Accepted
Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
This doesn't really require modern functional analytic tools, but we can prove a statement (due originally to Edelheit, according to Jochen Wengenroth in the comments) like
Let $V$ be a Frechet space,...
- 148k
14
votes
Is there a set of point $S \subset \mathbb R^2$ such that $|\{C: C \text{ is unit circle boundary }, |C \cap S| = 10\}| > |S|$
An alternative solution is the following:
Take 10 unit vectors in the generic position. Consider all $1024$ possible sums from the empty one (the zero vector) to the full sum of all $10$ vectors. ...
- 61.9k
13
votes
Accepted
Expected absolute value of the average of two points from the disc
The easiest way seems to be to take the integration variable to be $x= |\frac{z_1+z_2}{2}|$ and then integrate over the position of the point furthest from the origin:
This gives
$$ \operatorname{exp\...
- 3,927
12
votes
A variation of the law of large numbers for random points in a square
Interestingly, if we allow the lines to have arbitrary directions, it still requires roughly n^{4/3} (up to a log correction) lines to separate all the points.
https://www.cambridge.org/core/...
- 121
12
votes
Accepted
A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
I think the comment by @James Martin answers my question.
The number of clusters equals the number of pairs of points that are mutually nearest neighbors. The probability that a point is in a mutually ...
11
votes
Moments of area of random triangle inscribed in a circle
Let $A_d$ be the area of a triangle whose vertices are chosen uniformly at random from a unit sphere in $\mathbb{R}^d$.
I claim that for $d\ge 2$ and $m\ge 1$:
$$
E(A_d^{2m})=\frac{3}{4^m} \prod _{q=...
- 2,902
11
votes
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
Turning around the same ideas than previously exposed (I am not sure there exist two fundamentally different arguments), let me give a rough geometric proof.
Assume you project the uniform measure on ...
- 14.4k
11
votes
Accepted
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Both conjectures are true. The proof below uses fedja's method in a simplified form, and also a nice observation by Zacky (see the comments below this post).
Since $ax+by\geq c$ is equivalent to $b\...
- 105k
10
votes
Accepted
On 4 random points in a rectangle
As explained in Square Triangle Picking, the mean area of a triangle picked inside a rectangle of unit area is 11/144. So the probability that the fourth point lands inside this triangle is $11/144=0....
- 188k
10
votes
Taking points uniformly inside a general finite geometric domain
For an ellipse, one can rescale the coordinates so that the region becomes a disk and then sample in the way you mentioned.
However, in general sampling efficiently from irregular regions (or ...
- 6,001
8
votes
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?
Since you can scale everything linearly, let's suppose $\mathbb E(X) = \mathbb E(Y) = 1$.
I'm not sure this is optimal, but I think it must be close.
Consider a case
where all but two of the points ...
- 54.2k
8
votes
Marginal density of uniform spherical distribution
With $X$ uniformly distributed over the unit $n$-sphere, the joint probability distribution of all $n$ elements of $X$ is a Dirac delta function,
$$P(X_1,X_2,\ldots X_n)\propto\delta\left(1-\sum_{j=1}^...
- 188k
8
votes
A variation of the law of large numbers for random points in a square
Label your $N$ points as $(x_i,y_{\sigma(i)})$ with $x_1 < \cdots < x_N$ and $y_1 < \cdots < y_N$ ; this defines a uniform random permutation $\sigma \in \mathfrak{S}_N$, and all the ...
- 4,653
8
votes
Accepted
Local Lipschitzness of parameterization of Gaussians in Wasserstein space
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$The answer is yes.
Indeed, it is easy to see (cf. e.g. Proposition 7 or the beginning of its proof) that the Wasserstein distance between ...
- 128k
7
votes
The Largest Piece of Circumference
The distribution of the maximal distance between a pair of random points on the circle is known - when you scale it by $n/\log n$ you get a Gumbel distribution with scale 1, location 1., see, e.g.,
...
- 96.4k
7
votes
Definition of random measures
By way of introduction:
As expressed in some of the comments, I find the "locally compact" assumption possibly a bit too strong.
A weaker assumption than having a locally compact second-countable ...
- 698
7
votes
Accepted
What is the probability that a random chord in a sphere touches opposite hemispheres?
This is not a true "no pen or paper" solution requested by fedja, but at least it avoids integrals. :-)
Let $X$ and $Y$ be independent random vectors on the unit sphere. Write $E = (X - Y) / ...
- 17.2k
7
votes
Accepted
The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$
One can use basic probability theory to prove that
$$P(ab<kc)=\frac{2}{\pi}\arctan k,\qquad k>0.$$
Without loss of generality, the vertices opposite the sides $a,b,c$ are
$$A=e^{2i\beta},\qquad ...
- 105k
6
votes
Accepted
What is the nearest-neighbor distribution in this picture?
If $n$ points are placed uniformly at random in the unit square, then the distribution is very close to a Poisson process with intensity $n$. Scaling the process by $\sqrt n$, it’s like a Poisson ...
- 23.2k
6
votes
Accepted
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Without loss of generality, $R=1$.
Let $Z_1,\ldots,Z_n$ be iid standard normal random variables (r.v.'s). Then
\begin{equation}
\sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\...
- 128k
6
votes
Accepted
Show that the Markov chain of random tiling is irreducible
Suppose the sidelengths of your hexagon are $k, \ell, m$. A standard way of looking at this is to turn the picture by 30 degrees to the left and to view the yellow / cyan tiles as $m$ functions $f_i \...
- 10.3k
6
votes
A variation of the law of large numbers for random points in a square
This is to show rigorously that the uniform rectangular grid does not work -- cf. the answer by mike. As in the answer by Dieter Kadelka, suppose that the $cn$ vertical lines and the $cn$ horizontal ...
- 128k
6
votes
Taking points uniformly inside a general finite geometric domain
The approach (already alluded to in one of the answers) of sampling uniformly from a larger set and then throwing out the samples that you don't want is known as rejection sampling. You'll find all ...
- 47.4k
Only top scored, non community-wiki answers of a minimum length are eligible