# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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