1
vote
0answers
25 views

Slicing through a cuboid containing spheres, how many are exposed to the surface and what is their combined volume

So I place spheres of radius chosen at random from a normal distribution of known mean and standard deviation in a cub or cuboid at random (not overlapping) until a known density of the entire cube is ...
3
votes
2answers
83 views

Calculation for the chance of finding something a given distance from a starting point by walking straight in a random direction?

The premise is basically a 2D plane with a single point, the starting point. Now a landmark sought by a hiker is a certain distance from that point. If the hiker can only see 1 mile in any ...
6
votes
1answer
72 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
1
vote
1answer
45 views

Non-uniform sampling of N-sphere

Suppose I have a unit $N$-sphere from which I want to draw points at random. To obtain uniformly distributed points I do the usual technique of drawing $N$ random variables $x_i$ from a Gaussian ...
1
vote
1answer
21 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
0
votes
0answers
31 views

Geometric probability spaceships, Mars, communication, acute angles,

Let's assume that the shape of a certain planet is a ball. Three spaceships land independently on random points of the planet's surface. Every two of them can communicate directly by radio only if ...
0
votes
0answers
40 views

Find probability of angle being obtuse

We are given points A and B on the 2D plane and distance between them is 2. Let C - randomly picked point on the circle with radius R and center at the middle of AB. Find probability of angle ABC ...
2
votes
2answers
45 views

Probability of lies a point in a random triangle

We have a square. we will opt three random point from inside of this square and name it $p_{1},p_{2},p_{3}$ then opt another random point $p_{4}$. what is the probability of that $p_{4}$ lies in ...
5
votes
1answer
79 views

Expected time to completely cover a square with randomly placed smaller squares

Suppose I have the unit square $[0,1]^2$ and I choose a point $(x_1, y_1)$ randomly in a uniform manner inside $[0,1]^2$ and draw a filled in square of side length $1/N$ with center $(x_1, y_1)$. And ...
1
vote
1answer
73 views

Convolution of two Uniformly distributed r.v. ove

Assume a continuous random variable $X$ that is uniformly distributed $\underline{\text{on}}$ a $k$-sphere. For simplicity, lets assume a simple circle with radius $R$ in 2 dimension. Therefore ...
0
votes
0answers
19 views

Subdivision of Simplices

Consider a 1-simplex, a line segment; this can be subdivided into line segments all of equal area. Now consider the 2-simplex, an equilateral triangle; this also shares the property of being able to ...
2
votes
1answer
42 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
0
votes
0answers
36 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
0
votes
1answer
25 views

Furthermore fair dice?

In the field of board games, it is immediately apparent that fair die can be constructed for 2, 4, 6, 8, 10, 12 and 20 sides; represented by the coin, tetrahedron, cube, octahedron, decahedron, ...
2
votes
1answer
45 views

expected size of a special set of random points in the unit square

Today I came up with this fun problem, but I'm having a hard time to solve it completely myself. The question is the following: Let's generate n random points ...
0
votes
2answers
42 views

Explanation for the uniformity of the distance between a Gaussian variable to its nearest integer?

earlier I asked the question Expected distance for a gaussian variable to its nearest integer. and got a good answer. The expected distance is highly close to $1/4$, which is very similar to the ...
0
votes
1answer
35 views

probability for two vectors to lie on different regions created by hypeplane

Suppose we have two vectors $v_i,v_j$ and there is one hyperplane whose normal is chosen uniformly from the unit sphere. Then what will be the probability that $v_i$ lies on one side and $v_j$ lies on ...
0
votes
1answer
41 views

fixed length random chord outside of circle.

consider a uniform distribution on a unit circle, I construct a cord by the following steps: pick one endpoint A within the unit circle uniformly. points that are $0<d<1$ distance away from ...
2
votes
0answers
97 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
4
votes
1answer
67 views

Random embedding of $K_4$ in the unit square

Suppose I embed $K_4$ (the complete graph on 4 vertices) randomly in the unit square (using the uniform distribution for the positioning of the vertices). $K_4$ is planar, but not any embedding of it ...
1
vote
1answer
79 views

A point is selected uniformly at random in the interior of a unit square…

From It, the altitude to each side of the square is drawn. For each side, a stick of the altitude's length is obtained. Determine the probability that you can select three of the sticks and arrange ...
5
votes
3answers
285 views

find a chance that all N points lie on the half circle. [duplicate]

We are given a circle with N randomly allocated points on it. Task is to find a chance that all N points lie on the one half of circle. I have drafted some solution: 1. Since there are no way to put ...
2
votes
0answers
18 views

Distributions with a given mean and covariance

Fix $X := \mathbb R^d$ for some $d \ge 1$. Fix a vector $m \in X$ and a covariance operator $k : X^* \to X$, i.e., a symmetric, nonnegative-definite operator. Let $\Delta_{m,k}(X)$ denote the ...
4
votes
1answer
81 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
6
votes
1answer
117 views

Random Triangle Inscribed in a Circular Sector

Lately, I have been thinking about expected area and perimeter of a triangle inscribed in a 'partial' circle or circular sector with radius $r$ and truth be told, I couldn't answer these questions. I ...
2
votes
1answer
40 views

What's the chance for a dart to fall on a given line?

The Wikipedia article for "almost surely" gives this example. Suppose you throw a dart at a unit square. There is nothing else in the universe but you, the dart and the square - thus the dart must ...
0
votes
0answers
34 views

Predicting Likelihood of Uncertain Measurement

I am taking measurements of the distance from my position to a wall. The distance measurements contain error. The knowledge of the position of the wall contains uncertainty. Given knowledge of the ...
0
votes
0answers
69 views

Estimating the geometric shape of a point cloud without using the vertex information

Consider a point cloud format that describes 3D point clouds by vertices, triangle labels and normal vectors. If we miss the vertex information, is it possible to retrieve the lost data by triangle ...
1
vote
2answers
90 views

random circle with radius r on cartesian plane, probability of it not cutting x and y axis with intercepts.

I have a tough question here. Choose a circular disk of radius r on the cartesian plane. What's the probability it is not cut by horizontal lines with integer y intercept, or vertical lines with ...
2
votes
3answers
116 views

Random walk problem in the plane

Let a particle in the plane $R^2$ executes random jumps at discrete times $t= 1, 2, ...$. At each step, the particle jumps from the point it is a distance of lenght one. The angle of any new jump ...
5
votes
2answers
201 views

What is the average length of 2 points on a circle, with generalizations

I have earlier seen the question about finding the average length of two points and $n$ points inside the unit disk. But what about the more simple question, what happens if the points lie exactly on ...
2
votes
1answer
91 views

Expected value of maximum distance between points

Consider a two dimensional square domain ($S$) of size $l \times l$. We generate a point $\mathbf{x}_i = (x_i,y_i)$ in S with uniform distribution, i.e., the point is equally likely to be anywhere ...
2
votes
0answers
71 views

A follow-up to the regular hexagon question

This is a follow-up to the regular hexagon question. The problem statement was: Suppose we have a sphere and more than a half of its surface is red. Prove or disprove that we can place all ...
1
vote
1answer
103 views

Geometric Probability- Circle and two points

A point $P$ is chosen $0.5$ units away from the centre of a circle of diameter $2$. Now two points are chosen randomly on the circumference of the circle. What is the probability that the triangle ...
25
votes
3answers
502 views

Expected length of the shortest polygonal path connecting random points

$N$ points are selected in a uniformly distributed random way in a disk of a unit radius. Let $L(N)$ denote the expected length of the shortest polygonal path that visits each of the points at least ...
2
votes
0answers
68 views

Packing a larger sphere with smaller spheres in high dimensions

We were discussing today the probability of leaving a point uncovered while trying to fill a larger sphere by randomly throwing in smaller spheres. Here's the argument: We are working in ...
0
votes
1answer
197 views

Find the probability that splitting the unit interval into three random segments results in the sides of a triangle.

I found the question here. The precise problem is The unit interval is broken at two randomly chosen points along its length. Show that the probability that the lengths of the resulting three ...
2
votes
0answers
54 views

Expectation number of random points exactly on their convex hull

Suppose there are n random points uniformly distributed in a square, what's the expectation of the number of the points located exactly on the edge (or being vertexes) of their convex hull? What if ...
1
vote
0answers
133 views

A spaceship travelling to infinity while avoiding star collisions

Consider placing countably infinitely many points labeled $S_i$ randomly over $\mathbb{R}^2$, with asymptotic density points/area $ยต$. Then, what is the largest $r$ such that we can find a a ...
0
votes
1answer
264 views

Probability that coin will fall into a square

So the exercise is this: We have and infinite chessboard and we have a coin. Every grid is of length and width $a$, whereas the coin has diameter $2 \cdot r<a$. We throw a coin into a chessboard ...
1
vote
1answer
149 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 ...
12
votes
1answer
255 views

Random walks and diffusion limits

Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the ...
1
vote
0answers
37 views

an easy way to calculate d-volumes

If I have a function $f:\mathbb{R}^d\longrightarrow\mathbb{R}$ and I want to show that for all $[a_1,b_1]\times\cdots\times[a_d,b_d]\subset[0,1]^d$ the $f$-Volume is non-negative or non-positive, that ...
1
vote
2answers
274 views

Probability that centre of the square lies inside the circle joining the two points inside the square

Two points are uniformly and independently distributed (located) inside a square. A circle is drawn such that the segment joining the two points is a diameter. Find the probability that the center of ...
0
votes
0answers
37 views

Monte Carlo model of ultra loosely packed spheres

I have two questions. I am writing two monte carlo models that randomly propagate a square area and a cubic volume with mono sized hard disks and hard spheres. In the model the only criteria is that ...
1
vote
1answer
96 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 ...
0
votes
1answer
102 views

Griffths buffons needle

I've seen a lot of other proofs online about buffons needle, and I understand how they work, but I'm very confused about how Griffiths did this. I just can't visualize it. Needle Length L on ...
2
votes
0answers
40 views

getting PDF from a given Moment Generating Function

if the moment generating function mgf of a random variable w is M(t)=(1-7t)-20 find the i)pdf ii)mean iii)variance of w
1
vote
0answers
31 views

probabilistic location of points with normally distributed distances

I have a set of points (position-sensors) in 3D space attached to a structure which may be more or less rigid. At a given time each point may be in observed or unobserved state. My question is how to ...
5
votes
1answer
356 views

How to compute the expected distance to a nearest neighbor in an array of random vectors?

Let us have $k$ independent random vectors $x_1, x_2, \dots, x_k$ with uniform distribution over $ \left[0;1 \right]^n $. Then the distance (preferrably Manhattan) between an arbitrary vector $x_a$ ...