5
votes
2answers
49 views

How to characterize rotations in $\mathbb{R}^n$?

I am studying the performance of an optimizer algorithm to find the $$ \textrm{argmin}_{x\in \mathbb{R}^n} f(x) \text{ where } f : \mathbb{R}^n \rightarrow \mathbb{R} $$ I would like to test how the ...
0
votes
0answers
10 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
0
votes
1answer
24 views

probability calculation for position measurement being inside a circle

Consider a position measurement that is prone to a random error in any direction. This would mean that the position would be in a circle where the probability curve taken across the diameter would ...
1
vote
1answer
121 views

Determine if a set of points on a sphere come from a uniform distribution?

I have a large distribution of points on the unit sphere $S^2$ and I want to determine if those points came from a uniform distribution on the surface. Essentially, I'm looking for a two dimensional ...
1
vote
3answers
76 views

What should be the proportions of a three sided coin?

A classical coin has almost no chances of ending its course on the side when tossed. A round pencil with both ends flat has no chance of ending its course on the tip, when tossed. What would be the ...
1
vote
1answer
81 views

Probability using volumes wedge

Suppose that a point $(X, Y, Z)$ is chosen uniformly at random from the wedge $f(x ,y,z)$ belongs to $\mathbb{R}^3: 0 \leq x, y \leq 1, \textrm{and}\, 0 \leq z \leq x$. Compute the probability $((a ...
1
vote
1answer
43 views

Mean Distance on a 3-sphere?

What is the (analytical) mean geodesic distance between a set of randomly chosen points on a 3-sphere?
3
votes
2answers
107 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
8
votes
1answer
219 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two n-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
0
votes
1answer
64 views

Probability distribution of a random components vector

The vector $\rho$ has components: $(\rho_x,\rho_y,\rho_z)$ in a three - dimensional cartesian reference frame, where: $$\rho_x=a+\nu_x,\rho_y=b+\nu_y,\rho_z=c+\nu_z$$ with: $a,b,c$ constants and ...
5
votes
1answer
125 views

Probability of seeing to a certain distance in a forest and related problems

I was walking in a forest one day and saw trees all around me. I begun wondering about how far do I see in the forest on average. I was also reminded to the "proof" that the age of the universe is ...
4
votes
2answers
209 views

Joint density of the smallest and largest random variables among finite independent random variables with common density

I am trying to show the following result. Let $X_1, \ldots,X_n$ be independent random variables with the common density $f$ and distribution function $F$. If $X$ is the smallest and $Y$ the largest ...
2
votes
3answers
117 views

Which is the probability to a random line to be parallel to a specific other line?

In my perception, using the common sense, is less common, or less probable, to a random line be parallel that not to be, because to be parallel a line needs obey a restrictive rule. But anyone can, ...
4
votes
1answer
287 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
4
votes
3answers
2k views

Mean distance between 2 points within a sphere

I have found an answer on this site to the question of determining the mean straight-line distance between 2 randomly chosen points in a disc of radius r. I'm now trying to find an answer to the same ...
3
votes
0answers
103 views

Random points on a non-zero curvature 2D space

For some computational project, I'm interested in the pairwise distance matrix between random points on a unit square of $\mathbb{R}^2$. I now want to extend this case to non-zero curvature 2D ...
1
vote
2answers
1k views

Finding the mean distance between n points evenly distributed in a disc of radius r

In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light ...