All Questions
Tagged with pr.probability geometric-probability
164 questions
3
votes
1
answer
492
views
Random non-intersecting circles in the plane
If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
- 6,210
0
votes
0
answers
454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
1
vote
0
answers
234
views
Strong Dependence
I don't know if this definition has been already given.
Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. We say $Y$ is strongly dependent on $X$ if ...
- 1,109
2
votes
0
answers
242
views
What transformations preserve the von Mises distribution?
The von Mises distribution is entirely defined on the circle with a density given by
$$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$
where $x$ is in an arbitrary real interval of ...
CommunityBot
- 1
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 ...
- 28k
12
votes
2
answers
928
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 ...
CommunityBot
- 1
3
votes
2
answers
207
views
Connectivity of points sampled in a grid
Suppose that I partition an $n\times n$ square into $n^2$ squares $S_1 ,\dots, S_{n^2}$ each of area $1$, and then I sample a point $X_i$ uniformly at random in each $S_{i}$. Now fix a radius $r$ and ...
- 1,676
4
votes
2
answers
2k
views
Do Random Walks on the Hexagonal Lattice have a limit?
For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of $\mathbb{R}^...
- 3,069
3
votes
1
answer
528
views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
- 161
4
votes
1
answer
275
views
Nontrivial lower bounds on Cheeger inequalities for Markov chains
For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...
- 141
1
vote
0
answers
245
views
Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
CommunityBot
- 1
1
vote
1
answer
140
views
Equivalence between choosing a subspace and choosing its orthogonal
Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\...
- 60.5k
1
vote
1
answer
902
views
Product of densities of a wrapped normal distribution
The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...
CommunityBot
- 1
1
vote
0
answers
501
views
Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143