0
votes
2answers
30 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
26 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
24 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
73 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 $k$ ...
0
votes
1answer
20 views

Density on the square, expected value

Let $f: [0,1]^2 \rightarrow \Bbb R^{+}$ a density function on the square. I suppose that the random variable $X=(X_1,X_2)$ has the density f with respect to the lebesgue measure. I denote ...
4
votes
0answers
47 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
67 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
223 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 ...
1
vote
0answers
14 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 ...
1
vote
1answer
239 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
3
votes
1answer
60 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 ...
5
votes
1answer
90 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
34 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
28 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
53 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 ...
0
votes
2answers
64 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
106 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 ...
4
votes
2answers
121 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
71 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 ...
1
vote
0answers
64 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 ...
10
votes
1answer
68 views

Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
1
vote
0answers
68 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 ...
24
votes
3answers
478 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 ...
1
vote
0answers
52 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
136 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 ...
1
vote
0answers
40 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
130 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 $µ$. Then, what is the largest $r$ such that we can find a a ...
0
votes
1answer
151 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
120 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 ...
10
votes
1answer
234 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
35 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 ...
0
votes
2answers
233 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
32 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
145 views

random points in a cube

A point with coordinates $x$,$y$,$z$, is chosen uniformly at random from a cube: $$\{(x,y,z)\in \mathbb{R^3}:0\le x,y,z \le 10\}.$$ Assume that the probability of an event is proportional to 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 ...
0
votes
1answer
96 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
32 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
29 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
311 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 $x_a$ ...
1
vote
1answer
87 views

Expected minimum distance of a random point with respect a set of random points on the plane

I need to estimate, or bound, the expected minimum distance of a random point with respect to a set of other random points, all of which are located inside of a bounded rectangle. More specifically, ...
3
votes
1answer
1k views

Distance of a test point from the center of an ellipsoid

I'm trying to learn about Mahanalobis distance and I'm pretty close to getting the idea. I've learned that the distance has got a lot to do with the properties of an ellipsoid. I have understood so ...
0
votes
1answer
40 views

Probability that two points are more distant than a third equidistant point

Say you have three points $x,y,z \in \mathbf{R}^n$ with standard Euclidean distance $d$ and $d(x,y) = d(y,z)$. Then what's the probability that $d(x,z) > d(x,y)$ for random $x,z$? For convenience, ...
-1
votes
1answer
84 views

Probability that points are on a straight line

I am looking at a formula to calculate the probability that $n$ points are on a straight line between point $1$ and point $n$ in 2d Euclidean space. If the points are exactly on the line, the ...
1
vote
0answers
81 views

Expected number of pairs of intersecting chords

Suppose $ n $ chords are uniformly chosen on a circle. What will be the expected number of pairs of intersecting chords? From this discussion (Expected number of intersection points when $n$ random ...
1
vote
1answer
272 views

Probability that random vectors have a certain dot product

Given two random unit vectors in $\mathbb R^n$, one can consider the probability that the absolute value of their dot product is $x$, and thus form a probability density function. Supposing that the ...
1
vote
1answer
95 views

Can one sample uniformly from the surface of an $n$-sphere of non-unit radius using normal r.v.'s?

One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables ...
1
vote
0answers
110 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
4
votes
1answer
78 views

The Starry Rebound

An (infinitely small) ball starting out in the middle of a 5 pointed star table (outer 5 points 10m radius, inner 5 points 5m radius) has a starting angle of a random value from 0 to 360 degrees. The ...
0
votes
1answer
61 views

The probability distribution for the number of points in a randomly sampled area of a plane covered with $N$ points (placed with uniform probability)

With uniform probability, I place $N$ points on a plane of dimensions $D_x \times D_y$ and sample a region of area $A \leq D_x*D_y$ which has arbitrary geometry. What is the probability distribution ...
2
votes
1answer
572 views

Expected area of the intersection of two and three circles

We pick randomly two points, $p_1(x_1,y_1)$ and $p_2(x_2,y_2)$ inside a circle of origin $S$ with radius $R$ and we draw two circle $C_{1-2} (p_1,\sqrt {|x_1-x_2|²+|y_1-y_2|²})$ and $C_{2-1} ...