Probabilities of random geometric objects having certain properties (enclosing the origin, having an acute angle,...); expected counts, areas, ... of random geometric objects. For questions about the geometric distribution, use (probability-distributions) instead.

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13
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1answer
364 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
8
votes
1answer
176 views

A lawn, a flower, a pipe and the neighbours

You have a square lawn and a precious flower in the centre. You want to make sure you water the flower, and you don't particularly care how much of the lawn you water. To please your aleatory ...
6
votes
1answer
133 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 ...
4
votes
1answer
84 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 ...
3
votes
1answer
86 views

Expected length of minimum chord

You are given a circle of radius $1$. Suppose you pick $n$ independent points randomly on the circle and join neighboring points with lines to create chords. What is the expected length of the ...
2
votes
1answer
71 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 ...
2
votes
1answer
116 views

Convexity of a region on probability simplex

Exercise 2.15 g of Boyd et al Convex Optimization book : On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1,p_2,p_3,\ldots,p_n)$ corresponds to a distribution for random variable ...
1
vote
1answer
177 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 ...
0
votes
1answer
49 views

Geometric probabilities with rectangle

One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than ...
6
votes
0answers
181 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
4
votes
0answers
155 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
3
votes
0answers
47 views

Random convex shapes containing a ball

I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space. Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
2
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0answers
133 views

The Curvy Rebound: One of the most interesting (Geometric) Probability problems.

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also ...
2
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0answers
84 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 ...
2
votes
0answers
95 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
19 views

A sort of modified geometric series

I was wondering if there are any hints on how to manage this series $$ \sum_{i=2}^\infty \prod_{j=1}^{i-1} (1-cj^{\beta-1})=(1-c)+(1-c)(1-c2^{\beta-1})+(1-c)(1-c2^{\beta-1})(1-c3^{\beta-1})... $$ with ...
1
vote
0answers
32 views

Generate random numbers from appropriate distributions

Generate random numbers from appropriate distributions to find the area of the region enclosed by the curves y = sin (cos(x)), y = 0, x = pi/2 , and x = -pi/2 and report the area.
1
vote
0answers
77 views

Positivity of density for sum of dependent random variables

Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with ...
1
vote
0answers
44 views

Geometric probability with square

Jhon and Simon have common bank account which has $720$ dollars. Each of them has to buy a gift independently from other (Gift cost $< 360$ dol). What is the probability that after shopping there ...
1
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0answers
24 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
1
vote
0answers
66 views

Probability of a coin falling on the edges of a square

Let a coin be randomly (and uniformly) dropped onto a square on the floor. Assume the edge length of the square to be $ d $ and the radius of the coin to be $ r < d/4$. I know that the probability ...
1
vote
0answers
40 views

Shortest path length when edge length is limited

$N$ nodes are uniformly distributed in a square whose side length is $1$. There exists an undirected edge between two nodes, if and only if the distance between them is less than or equal to $r$. Here ...
0
votes
0answers
26 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
0
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0answers
18 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
0
votes
0answers
16 views

Inequalities for Laguerre polynomials

The following inequality holds, $$ \Big( 4\int_0^\infty rdr \big|\mathcal{L}_1(4r^2)\big|e^{-2r^2}\Big)^3 \geq 4\int_0^\infty rdr \big|\mathcal{L}_3(4r^2)\big|e^{-2r^2}, $$ where $\mathcal{L}_n(x)$ ...
0
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0answers
28 views

Buffon noodle problem gives theoretical issues

Let $\Gamma$ be a rectifiable curve in plane, having length $l$. Denote by $X_{\Gamma}$ the random variable that represents the number of crossings between $\Gamma$ and a grid of $d$-spaced parallel ...
0
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0answers
26 views

What is the probability of uniformly sampling a point in d-dimensional hypercube?

Let us consider a hyper-cube whose length is l units along each of its d-dimensional structure. It is desired to uniformly sample a point inside the hyper-cube. How to do uniform sampling and what ...
0
votes
0answers
41 views

How long does one remain in the square

This is a slightly known question: There is a square of unit side. Its centre is $O$. Two movable points $X$ and $Y$ are placed randomly in the square. Let $A$ be the midpoint of $OX$ and $B$ ...
0
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0answers
17 views

Is there a way of solving this probability question without using the survival characteristic of a geometric random variable?

I have the following problem, and the author presents the solution by using the survival characteristic of a geometric random variable: However, I am not very familiar with this method and was ...
0
votes
0answers
19 views

How is the Dehn invariant related to the mean width?

Reading Ravi Vakils Monthly article of february 2011 and watching the video; he mentions that the Dehn invariant is related to the linear invariant measure $\mu_1$ of geometric probability. The Klain ...
0
votes
0answers
27 views

The expected number of polygons created as a result of the intersection between randomly placed rectangles inside a square

How can I compute the expected number of polygons created as a result of the intersection of $k$ rectangles of area $B$ each, which fall randomly inside a square of area A? Regarding how randomness ...