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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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Expected area of triangle inscribed in a circle

On a unit circle , BC is a chord with length 4/π . Point A is picked randomly at its circumference . Find the expected area of △ ABC .
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generalization of Dyck Path: size K upward steps

One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 ...
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Probability of two random points being orthogonal in higher-dimensional unit sphere

I understand that most points will be close to surface due to volume concentration. Also I also understand the concentration of volume near the equator, relative to any specific point (North pole). ...
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Expected distance between two points on randomly selected line segment

My question is similar to a question answered before many times, namely the expected distance of L/3 between two random points on a line segment (Average Distance Between Random Points on a Line ...
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Answer To A Probability Problem About Placing Random Points In A Circle.

Problem Statement: Three dots are randomly placed in a circle of radius one cm. What is the probability that when a fourth dot is placed (randomly) in the circle, it is at least one cm away any of the ...
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Continuity of the Euler characteristic with respect to the Hausdorff metric

Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means ...
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Expected value of the number of steps in a walk around the rectangle.

We have a rectangle $ABCD$. The walk starts at $A$. Let $X$ be the time required to visit all the vertexes. (We independently go to the adjacent vertex with probability 1/2). Calculate $EX$. Some ...
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40 views

Expectation problem with geometric random variable?

If I have a stream of integers from 0 to 9(each with equal probability and repetition is allowed) that are spewed out until I get the first instance of 4-5 next to each other in this order, how do I ...
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$\min\{X, Y\}$ is geometrically distributed according to parameter $1 - (1-p)^{2}$

Let $X,Y$ be independent from the parameter $0 < p < 1$ distributed random variable on a discrete probability space $(\Omega, \mathcal{F}, \mathbb P)$. Show that $\min\{X,Y\}$ is geometrically ...
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Voronoi cell volume inside the ball

I have the following problem: Let us denote a ball with center $C$ and radius $R$ in $\mathbb{R}^d$ as $B(C, R)$. Given a unit ball $B(0, 1)$ and vector $u$ has a uniform distribution inside the ...
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Find probability that a line l may be tangent to circle $x^2+y^2=n^2\left(1-(1-\frac{1}{\sqrt n})^2\right)$

Consider the set $A_{n}$ of points $(x,y)$ where $0\leq x\leq n,0\leq y\leq n$ where $x,y,n$ are integers. Let $S_{n}$ be the set of all lines passing through at least two distinct points of $A_{n}$. ...
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Expected minimum absolute value of unifrom distribution

Consider we draw N iid values $x_1, x_2,...,x_N$ from an uniform distribution on $[- \epsilon, \epsilon]$. Can we calculate the expected minimum of their absolute values: $$ \mathrm{E}(\mathrm{min}(|...
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Nearest distance distribution from a node

Please see the scenario below. I am interested in finding the probability density function of green point to its nearest blue point in circle with Radius Rc. This must not include any point within ...
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Find the probability that A,B,C are connected

I was given the following problem as a homework assignment: Denote with $S$ the ball with center $O$. Three points $A, B$ and $C$ are chosen at random on its surface, their positions being ...
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1answer
107 views

Expected length of arc containing a point

You are given a circle $C$ of unit radius and a point $M$ on it. Now, three points are chosen at random on the circle $C$ which divides it into three arcs. What is the expected length of the part that ...
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Probability that the triangle is acute

A triangle is formed by randomly choosing three distinct points on the circumference of a circle and joining them. What is the probability that the formed triangle is an acute triangle?
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CDF and Survival Function of Geometric Distribution

I am having trouble understanding the intuition behind the CDF and survival probability of a geometric distribution on both {0, 1, ...} and on {1, 2, 3, ...}. I know that a geometric starting from 0 ...
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What is the probability that coin will not touch lines of segments?

In geometric probability I want to know what is the probability that a small coin (r < a) will not touch the lines of segments with 2*a distance(I mean the plain is divided into 2*a length segments ...
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Probability of joint distribution has closed form answer?

Let $Y$ be a geometric distribution with a parameter $\theta$ such that $0 < \theta < 1$ and let also $P(X=x|Y=y)$ be a binomial distribution with parameters "y" and "1/2". So: $$ P(X=x|Y=y) = ...
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A geometric probability question

Find the probability of distance of two points ,which are selected in $[0,a]$ closed interval, is less than $ka$ $k \lt 1$ What did I write : $P(A)$ = (Area measure of set $A$)/(Area measure of set $...
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What is the surface area an ellipse takes up on a sphere?

Take $\Sigma$ a $k\times k$ positive-definite real matrix and $E$ to be an associated ellipse: $$E:=\{(x_1,\dots, x_N): \frac{1}{N}\sum_n x_n^\dagger \Sigma x_n \leq 1\}.$$ Now take $z$ uniform on $\...
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PDF Modification from a uniform to a way-point spatial distribution

Basically, I know that for a uniformly distributed points, r within the intersection $XQYD$ in Figure 1, the pdf is given by $$f(r) = \frac{l(r)}{A}$$ where A is the area of the intersection region ...
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Conditioning a PDF in the intersection of two circles.

Given the diagram below (The first figure): $$f_r(r)= \frac{\lambda \pi r \exp{(-\lambda \pi r^2/2)}}{1-e^{-\lambda \pi R^2/2}}$$ where $$0<r\leq R$$ The pdf $$f_{\text{Z}}(\text{z})=\frac{2\...
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1answer
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Geometric probability - challenging problem(two points of a square K determine a diagonal of another square that is contained in given square K)

Let $K:=[0,1]^{2}$ be a square on $\mathbb{R}^{2}$. We select 2 random points $A$, $B$ $\in [0,1]^{2}$ in this square. What is the probability that the square, whose diagonal is the line segment $AB$, ...
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What is the probability that 3 random points in $2n$ regular polygon contains the center of gravity

What is probability that $3$ random point in $2n$ reguler polygon contains center of gravity I contain the diagram to make up my explanation. Left one is contain the center of gravity,while right ...
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1answer
61 views

Probability that the distance between two random points inside a circle is less than some value

Two points are chosen at random within a circle of radius R. What is the probability that the distance between them is less than D? I can solve the problem for the special case $R=D$ as follows: For ...
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Bounding probability of picking point from $l^p$ ball [closed]

Consider picking $x$ randomly and uniformly from the $l^p$ ball with radius $R$, $B_p^d(R)$. If $0\le \gamma \lt 1$, how can I conclude following? $$ P(\| x\|_p \le \gamma R) = \frac{\...
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1answer
24 views

Geometric distribution random

In reading a research paper I came across this note, I am not sure I fully understand it. I am wondering if they are referring to P as the geometric distribution so their output would be $p^r$ where $...
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Probability of finding nodes in a moving circle

This seems to be a simple problem to me, but am not getting right results. As attached in the figure (not drawn to scale). A circle centered at $X$ (X in Blue) with initial position at time $t=t_0$ ...
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1answer
49 views

Difference of numbers in a unit interval

$x,y,z\in \mathbb{R}$ are chosen at random from the unit interval $[0, 1]$. What is the probability that $$\max(x,y,z) - \min(x,y,z) \leq \frac{2}{3}$$ EDIT- Solutions not using calculus would be ...
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Probability Question Help- Statistics, Distribution, Geometric Distribution and Normalization.

I am having trouble with a probability question regarding to G-distribution. Background: Let's assume I have a disc that has two sides, one is black and one is white. Note: the disc has biased ...
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Interview Question on Probability: A and B toss a dice with 1 to n faces in an alternative way

A and B toss a dice with 1 to n faces in an alternative way, the game is over when a face shows up with point less than the previous toss and that person loses. What is the probability of the first ...
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187 views

Average distance detween two random points on two line segments

Suppose you have two straight line of length $L_1$ and $L_2$, and a point is chosen at random along each line. What is the expected distance between these points? This question is a complement of ...
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Probability that a random graph will remain planar after adding an edge

According to this answer, a random graph on $n$ vertices is a graph which has each of the $n\choose2$ edges independently with probability $1/2$ each. The probability of at most $3n-6$ edges (which is ...
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Broken stick probability question (variation)

A stick is broken into two at random, then the longer half is broken again into two pieces at random. What is the probability that the three pieces make a triangle? I've been stumped at this question ...
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Valuation Property for mean width

For some polyhedron, $P$, define the mean width function, $$H(P)=\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$ Where $E$ is the set of all edges of the polyhedron, $L_e$ is the length of edge $e$ and $\...
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How to compute the expected distance to the nearest neighbor of higher density (normal distribution)?

Suppose we have $X_1, \dots, X_n$ independent random variables with distribution $N(0, 1)$. The density function is: \begin{equation} \phi(x) = \frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}} \end{equation}...
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Cumulative distribution function for geometric random variable

For geometric random variable $f(k)=(1-p)^{k-1}p$. Then $$F(k) = P(X\leq x)= 1-P(x>k) = 1 - \sum_{i=k+1}p(1-p)^{i-1} = 1 - (1-p)^k \sum_{i=1}^ \infty p(1-p)^{i-1}= 1 - (1-p)^k$$ I would ...
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Estimate the volume of Voronoi cell

Let given a ball of radius $\alpha$ centered in point $u$ in $d$-dimensional space. Let given a sample of $n$ uniformly distributed vectors $x_i$ ($i = 1,\dots,n$) inside the ball. For each vector $...
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1answer
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CDF for distance between two points on line segment

I am trying to solve a paper and have a basic question which I am unable to understand. Given a line segment $[0,a]$, two points $x,y$ are randomly selected such that $\mathcal{D} = |x-y| \leq l$. So ...
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1answer
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Probability that a stick randomly broken in three places can form a triangle

I like questions about geometric probability, and two of my favourite questions here on math.SE are Probability that a stick randomly broken in two places can form a triangle and Probability that a ...
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218 views

Intersection of random line segments in the plane

Let a point on the plane be randomly chosen via $(\sqrt{\frac{t}{1-t}}\cos(2\pi\theta),\sqrt{\frac{t}{1-t}}\sin(2\pi\theta))$, where $t$ and $\theta$ are uniformly randomly chosen on $[0,1]$ (...
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Probability in geometry: finding the probability of selecting a $5$-tuple such that a pentagon can be formed from its sides

Consider all tuples of $5$ real numbers each less than $5$. Find the probability of selecting a tuple such that a pentagon could be formed from its sides. My attempt : I tried to visualize the ...
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Probability of forming a triangle of given perimeter. [duplicate]

Given a perimeter P of a triangle, three lengths which are real number are chosen to have sum as P. What is the probability that triangle can be formed by those 3 lengths?
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Intersection of random line segments on the sphere

This is actually two questions, that have a similar premise: If points in $\mathbb{R}^2$ are chosen stereographically randomly (i.e. chosen uniformly randomly on the surface of the unit sphere and ...
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Lower-bounding Gaussian inner products with high probability

Suppose that $K\subseteq \mathbb R^n$ is a proper convex set with piecewise smooth boundary and that $0 \in K$. Assume that $x \in K$ and let $z \sim \mathcal{N}(0, I_n)$ be a Gaussian random vector. ...
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Probability that a delaunay triangle contains the center of its circumcircle

A Delaunay triangulation for a given set P of discrete points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). https://en.wikipedia.org/...
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Expectation of the distance to the center and to the boundary of a point in a circle

We randomly choose a point inside a circle of radius 1. Let $X$ be the distance of the point to the center of the circle and $Y$ be the distance of the point to the circle boundary. What is the ...
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Hands-On Matlab Resources for Wireless Networks Modeling using Stochastic Geometry

Stochastic Geometry has become a very strong mathematical tool for studying and understanding several aspects of wireless communication and networks. As I write this, I find quite a large number of ...
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How to derive the pdf of some geometric quantity

I would like to know how one cap derive the pdf of some geometric quantity. My problem goes as follows: Consider the hypersphere $\mathbb{S}^{n-1}$ (hypersphere in $n$ dimensions) and randomly choose ...