Questions tagged [geometric-probability]

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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33 views

For what kind of random vectors do we have $\sup_{p \ge 1}\|X\|_p < \infty$?

Let $X$ be a random vector on $\mathbb R^m$ (assumed to have zero mean, for simplicity). For $p \in [1,\infty)$, define $e_p(X):=\mathbb E\sum_{j=1}^m|X_j|^p \in [0,\infty]$. Finally, define $\|X\|_p \...
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1answer
44 views

The Broken Stick Problem (Please check my approach) [duplicate]

There is this famous probability problem called the broken stick problem. The problem is: If a stick of length x is broken into three pieces, what is the probability that the three pieces can be used ...
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Explicit tail bound for coordinates of uniform random vector on euclidean ball in high-dimensions

Let $X$ be drawn uniformly at random from a euclidean ball in $\mathbb R^n$ (the dimensionality $n$ is large!) around the origin and of finite radius, and let $M$ be the median of $|X_1|$. In Super-...
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If $X$ is a nonnegative $\sigma$-subGaussian random variable with $P(X=0)\ge p$, what is a good upper bound for $P(X \ge h)$?

Let $X$ be a nonnegative random variable and let $\sigma \in [0,\infty)$ and $p \in (0,1)$ such that (1) $P(X=0) \ge p$ (2) $Var(X) \le \sigma^2$ For $h \ge 0$, define $c_X(h):=P(X \ge h)$. The ...
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54 views

Upper bound on intersection of $\ell_p$-norm balls in $\mathbb R^n$, with different centers and the same radius

Let $x$ and $x'$ be points in $\mathbb R^n$ (for a large positive integer $n$) and let $d(x,x') := \sup_{1 \le j \le n} |x_j-x'_j|$ be their $\ell_\infty$-norm distance apart. Given $r \ge 0$, let $B(...
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Is there an analytic expression for the geometric (Fréchet) mean of the distance between $n$ points in a two-dimensional Euclidean space?

I am trying to find out whether there is a convenient probability distribution over the Euclidean plane such that the geometric mean (or more generally the Fréchet mean) of the pairwise distance ...
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58 views

Min distance from a random point in disk to $n$ i.i.d random points in the same disk

Pick a point $A$ uniformly within a disk of radius 1. Then pick another $n>1$ points $P_1, P_2,..., P_n$ uniformly from the same disk. What is the expected value of $\min\{AP_1, AP_2, ..., AP_n\}$? ...
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Roulette probabilities “equality”

If a roulette has equal number of two differently coloured "boxes" arranged alternatingly, the probability of a ball landing on either of the colours should be 1/2. If the like coloured "boxes" were ...
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How to find the average distance between two points in an $n$-sided polygon?

I just saw a few videos and read some answers on this site regarding the average distance between two points in a circle and a square, but I was wondering if we can do so for an $n$-sided regular ...
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Concentration: lower bound for $\sup_u \mu(\{a + \epsilon u \mid a \in A\})$, where the sup is over unit vectors $u \in \mathbb R^n$

Let $\mu$ be a probability distribution on $\mathbb R^n$. For $\epsilon > 0$ and a Borell set $A \subseteq R^n$ with $\mu(A) > 0$, define the $\epsilon$-neighborhood of $A$ as $A^\epsilon := \{...
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For $ x,y,z \in (0,1) $ chosen randomly with uniform distribution, what's the probability that $x+y+z<1$? [duplicate]

I tried the $2D$ case with $x,y \in (0,1)$ and $P(x+y < 1) = \frac{1}{2}$ I got this by sketching the inequalities in the question, namely $ 0\leq x,y\leq 1, $ and $y < 1-x$ and seeing that ...
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Given $372$ points in a circle with a radius of $10$, there is an annulus with radii $2$ and $3$ containing at least $12$ of these points.

Given $372$ points in a circle with a radius of $10$, show that there is an annulus with inner radius $2$ and outer radius $3$, which contains not less than $12$ of the given points. My thinking is ...
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Probability that a fixed point on a table of radius R and border of width r will be covered given N circular disks of radius r staying on the table

Circular discs of radius $r$ are thrown at random on to a plane circular table of radius $R$ which is surrounded by a border of uniform width $r$ lying in the same plane as the table. If the discs are ...
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1answer
32 views

Count of random points belonging to a Voronoi cell in the unit square

I am considering a Voronoi tessellation in the unit square $V \in [0,1]^2$ for $G$ points uniformly randomly distributed. Then I am considering $N$ other points also randomly distributed uniformly in $...
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1answer
34 views

What is the average length of threads criss-crossing a hollow sphere?

Imagine a hollow sphere of radius $R$ that has a large, random (but even) number of holes in it. The surface density of the holes is constant. Threads criss-cross the sphere at random, from one hole ...
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Computation of joint distribution of two random variables on the unit disk

Consider the probability density function on the 2-D unit disk $\{(x,y)\in \mathbb{R}^2: x^2 + y^2=1\}$ given by $f(x,y) = \frac{3}{2\pi} \sqrt{x^2+y^2}$ I would like to compute the joint ...
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Convergence to an $\ell_p$ ball, of Steiner symmetrization of compact convex subsets of $\mathbb R^n$

Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball... Fix $p \in [1,\...
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80 views

Number of random unit vectors which are less than theta apart

Given $n$ unit vectors which are uniformly distributed on a unit sphere, what the expected number of groups of $k$ vectors which are within an angle $\theta$ of one another For example, if I have $n=...
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55 views

Density of the first $k$ coordinates of a uniform random variable

Suppose that $X$ is distributed uniformly in the $n$-sphere $\sqrt{n}\mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ coordinates ...
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70 views

Prove that probability of choosing an isosceles traingle in Set of traingles is $0$.

$S$ is set of triangles of unit area. All members of $S$ are uniformly distributed. Let $A$ be the event that a randomly chosen member of $S$ is an isosceles triangle. Prove that the probability of $A$...
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46 views

Probability two splits make triangle

This is a slight variation on the usual broken stick problem. A stick is broken randomly into two pieces. The larger piece is then broken in two. What is the probability the pieces can form a ...
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53 views

Finite set of vectors approximating a unit ball.

I am having difficulty proving that a unit ball can be approximated with a set of finite vectors. Specifically, I want to bound the error of the following approximation. Let $D$ be a uniform ...
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25 views

Given $A \in \mathbb R^{m \times n}$, find upper bound for $\mathbb E\|Az\|_q$ for $z$ drawn uniformly at random on the sphere $\{\|z\|_p = 1\}$

Let $m$ and $n$ be positive integers and $p,q \in [1,\infty]$. Consider the finite-dimensiaonal normed vector spaces $X = (\mathbb R^m,\|\cdot\|_p)$ and $Y = (\mathbb R^n,\|\cdot\|_q)$, where $$ \|x\|...
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28 views

Expected number of hyperplane cuts to partition a $d$-dimensional hypercube of side length $n$ into pieces of unit volume

I came up with the following question and am not certain of its solution. Has this problem or problems like it be tackled before? Let $H_d$ be the $d$-dimensional hypercube $[0,n]^d \subset \mathbb{...
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3answers
56 views

Find the probability that a stick will lie entirely on the tile.

A floor is paved with tiles, each tile being a parallelogram such that the distance between pairs of opposite sides are $a$ and $b$ respectively, the length of diagonal being $L$. A stick of length $C$...
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2answers
51 views

Average angle between two randomly chosen vectors in a unit square

Consider two randomly chosen vectors $<a,b>$ and $<c, d>$ within the unit square, where $a, b, c,$ and $d$ are chosen uniformly from $[0,1]$. What is the expected angle between the vectors?...
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Show that the probability that it falls entirely on one brick is $\frac{(a-c)(b-c)}{ab},( c<a, c<b).$ [duplicate]

Question: A floor is paved with rectangular bricks each of length $a$ units and breadth $b$ units. A circular disc of diameter c is thrown on the floor. Show that the probability that it falls ...
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1answer
38 views

how close can you get to a random d-dimensional vector of +1 and -1 given k guesses?

Consider a uniformly randomly selected vector $v \in \lbrace +1,-1 \rbrace^d $ that is a vector of size d, consisting of +1 and -1 (there are 2^d such vectors) I'm interested in understanding, how "...
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2answers
52 views

Geometric probability — line segment [closed]

Fellow math lovers! It's been quite sometime since I have solved basic probability problems. I am now trying to remember how to calculate geometric probability. As far as I remember, the general ...
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1answer
52 views

What is the probability that the larger of two independent uniform variables on $[0,1]$ is greater than $3/4$ if the smaller one is less than $1/4$?

Two independent random variables are uniformly distributed on $[0, 1]$. The question asks if the smaller of the two numbers is strictly less than $\frac{1}{4}$, then what is the probability that ...
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1answer
28 views

Geometric probability; circle and two points

On a circumference of circle which radius is 1, two points are chosen randomly. What is the probability that the distance between these two points is less then 1? The solution from my book is $\frac{...
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Derivation of Discrete Phase Type Distribution

For an independent project I am trying to derive the cumulative function of the discrete phase type distribution. I was able to obtain the mass function: $f(x) = \alpha T^{(x-1)}t$ where $\alpha$ is ...
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How to compute the expected value of the number of domains of a given size?

Given a regular lattice (i.e., a finite set of identical squares - or triangles, or hexagons - stacked next to each other as a large rectangle - or some other shape), each border between cells is ...
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Geometric probability; circle doesn't intersect with lines

On a plane covered with parallel lines, which distances are alternately 2 and 3, circle with diameter 1 is thrown. What is the probability that the circle doesn't intersect with any of the lines? If ...
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1answer
78 views

On a certain discrepancy measure between probability distributions on the symmetric group of permutation $\mathfrak S_n$

Let $\mathfrak S_n$ be the symmetric group of permutations on $n$ objects and let $P$ and $Q$ be a probability distributions on $\mathfrak S_n$ (i.e $P$ and $Q$ are points on the $n!$-simplex). For $1 ...
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341 views

Probability Based on a Grid of Lights

The question is as follows : A grid of $n\times n$ ($n\ge 3$) lights is connected to a switch in such a way that each light has a $50\%$ chance of lighting up when switched on. What is the probability ...
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1answer
82 views

Average distance of two points on a circle

I stumbled upon the question of the average distance of two points on a circle, I learned how to calculate this with polar cordinates (find the distance as a function of $\theta$, itegrate the general ...
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Compute the variance of the random variable X measuring the distance between P1 and P2 [duplicate]

Can anyone help me with this? Consider the circle $C = \{f(x,y)\in\mathbb{R^2}: x^2 + y^2 = r^2\}$ and a point P2\in\mathbb{C}. A point P2\in\mathbb{C} is randomly chosen. Compute the variance of ...
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Probability of Constructing a Triangle From Line Segments of the Interval $[0, 1]$ [duplicate]

If you randomly select two values $X$ and $Y$ in the interval $[0, 1]$, it divides up the interval into 3 line segments. What would be the probability that a triangle can be constructed from these ...
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A random sphere containing the center of the unit cube

Inspired by a Putnam problem, I came up with the following question: A point in randomly chosen in the unit cube, a sphere is then created using the random point as the center such that the sphere ...
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What is the expected volume of the simplex formed by $n+1$ points independently uniformly distributed on $\mathbb S^{n-1}$?

I was surprised that I couldn’t find this question answered on this site (not for lack of trying). I need the result for answering Probability of random sphere lying inside the unit ball, so I&...
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417 views

Probability of random sphere lying inside the unit ball

Let $n\geq2$. Let $B\subseteq\mathbb{R}^n$ be the unit ball. Randomly choose $n+1$ points of $B$ (uniformly and independently). Then (almost surely) there will be a unique hypersphere $S$ passing ...
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Definition of Random Measures

Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from? I would like to have a motivation ...
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1answer
46 views

Mathematical expectation of the average distance of some set of points

In cartesian plane, $N$ points are randomly generated within a unit square defined by points $(0,0) (0,1) (1,1) (1,0)$, with uniform distribution. What is the mathematical expectation of the average ...
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350 views

Expected tetrahedron volume from normal distribution

Two equivalent formulas for the volume of a random tetrahedron are given. Further on you can find an interesting conjecture for the expected volume that shall be proved. Tetrahedron volume Given are ...
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Probability for individual elements

I have Probability equation given by $p(r)=\frac{ (1-\alpha) \alpha^{CW}} { 1- \alpha^{CW} }. \alpha^{-r} $ . Here r is number of objects from 1 to $CW$. How can i write probability of picking any ...
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2answers
188 views

Uniform random distribution on a unit disk

a) A point is uniformly chosen in the unit disk $0 ≤ x^2 + y^2 ≤ 1$. Find the probability that its distance from the origin is less than $r$, for $0 ≤ r ≤ 1$. b) Compute its expected distance ...
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1answer
52 views

Is sum of expected triangle areas equal to expected area of triangle sums?

Starting point Given are 4 multinormal distributions $\mathcal{N}(\vec{\mu}_1,\Sigma), \mathcal{N}(\vec{\mu}_2,\Sigma),\mathcal{N}(\vec{\mu}_3,\Sigma),\mathcal{N}(\vec{\mu}_4,\Sigma)$ in $\mathbb{R^3}$...
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2answers
128 views

Point picking in a 1x1 square: probability of line segments connecting 2 random interior points to catacorner vertices intersecting in the square.

Like many of the best problems I found this one on twitter. "In a square with side length 1, two random points in the square are connected by segments to two opposite vertices. How likely is it that ...
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1answer
57 views

Does expected triangle area change if a random point is added?

Starting point Case I There are 3 random points in a volume. Calculate the expected area of the triangle. Case II Calculate the expected area of any of the 4 triangles that are formed if a 4th random ...

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