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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Calculate expected number of heads in 10+$\xi$ coin tosses (GRE problem)
This is a problem from a preparatory GRE preparatory GRE test made by guys form University of Chicago.
Problem:
A man flips $10$ coins. With $H$ the number of heads, and $T$ the number of tails, the ...
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Geometric probability: Estimating regions devoid of random points
First, generate $N$ random points in $(0,1)^2$ according to the standard uniform distribution $\mathrm{U}(0,1).$ Then generate and superpose a set of $S$ curves $\phi_S(x)=\exp\bigg(\frac{\log^2 S}{\...
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What is the probability that the distance between $a$ and $b$ is greater than $3$?
Points $(x,y)$ are selected at random where $0\leq x\leq3$ and $−2\leq y \leq 0$. This means that for instance, the chance $(x,y)$ belongs to the square $[1/2,2]\times[−1,0]$ equals $\frac{1.5}{6}$.
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Bounding variables $X$ and $X + 1$ using Markov's inequality
Suppose $X \sim \text{Geometric}\left(\frac{1}{2}\right)$ where $X$ is the number of failures. Then let $Y = X + 1$ (i.e., $Y$ counts the number of trials). Then we have,
$$E(X) = \frac{1}{p} - 1 = 2 -...
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Singular values of uniform random points on hypersphere?
This question is motivated by a self-supervised learning problem in machine learning, but I'll try to strip out as many unnecessary details as possible. In this setting, we have large datasets and we ...
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Expected angle between random directions in $\mathbb{R}^2$
On page 50, line 2 of the High Dimensional Probability book (HDP-book.pdf), the author asserts that the expected angle between 2 random directions is $\pi/4$, however, I keep ending up with $\pi/2$. ...
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Distances of points in unit square to N random anchor points
Assume that N points are distributed uniformly randomly within the unit square, and call these points anchor points.
What is the mean, mean minimum, and mean maximum distance of a point randomly ...
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What is the probability a Quadrilateral formed from a circle has an area of over half the circle
The question is :
Pick 4 random points along the circumference of a unit circle, 1 in each quadrant labeled a, b, c, and d respective to the 1st, 2nd, 3rd, and 4th quadrants. Connect these points to ...
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Break a stick at $n$ random points. What is the probability that the three shortest pieces can form a triangle, as $n\to\infty$?
Question :
On a stick, choose $n$ uniformly random points, and break the stick at those points. What is the limit of the probability that the three shortest pieces can form a triangle, as $n\to\infty$?...
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A geometrical puzzle involving calculus
Some time ago I stumbled across a problem from the Putnam Mathematical Competition. I could not find it, but I remember the text quite well.
There are two vectors: a=(10, $y$) and b=($x$,10), where $0 ...
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Probability function for distance to the nearest neighbour given $n$ points distributed randomly on a line? [closed]
Note: I'm not familiar with a lot of mathematical terminology, so please excuse any misuses. I'd also make a request to use simpler language (if possible) in an answer
What I'm seeking is similar to ...
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Expected norm-squared of one random vector projected onto others, all iid
For $k\leq n$, $x_1,\dots,x_k \in \mathbb{C}^n$ are independent identically distributed random vectors almost surely unit norm and with span dimension $k$. Call $X=[x_2,\dots,x_k]$. I am studying $\...
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Draw a handful of random vectors iid. Is projecting one onto the handful essentially the same as projecting one onto another?
Fix some distribution $D$ over the unit sphere in $\mathbb{C}^n$.
For $k<n$ and $x_0,x_1,\dots,x_k \overset{iid}\sim D$, call $X=[x_1,\dots,x_k]$ and identify the projection onto $\operatorname{...
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Uniform Distribution of Chords
In the context of the Bertrand Paradox, I understand that different methods of choosing a chord lead to different probability density functions. For instance, selecting a chord based on a random ...
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Distribution of lines given by choosing two random points on the sphere according to distribution P
The problem:
Given some distribution $P(x, y, z)$ over the surface of the sphere $S^2$, what is the distribution of lines $L$ generated by choosing two points $p_1, p_2 \sim P$ and returning the line $...
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80% favors the building of a leisure center. Find the probability that the 7th Person is the 2nd person who is not in favor of the leisure centre.
For this question, I know I need to use Geometric Distribution. But I am lost because question asked for the 2nd person i.e 2nd occurrence instead of 1st occurrence. Appreciate anyone's help. Thanks!
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Distribution of number of points in a region of blue noise
Let $X$ be a spatial blue noise point process in the plane where points are at least distance $d$ apart for $d > 0$.
For a closed region in the plane $B$, let $N(B)$ count the number of points in $...
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Probability that a line intersect two other lines inside the unit disk.
Consider the unit disk: $D=\{(x,y)\in\mathbb{R}^{2}:x^2+y^2\leq 1\}$ and the lines $x=0$ and $y=0$. We want to find the probability to draw a line which intersect the first two with intersection ...
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Probability that the Origin Lies in the Convex Hull of Non-identically Distributed Random Vectors
Suppose that we have a set of random vectors $\mathcal{A}=\{A_1,\dots,A_m\}$, where each $A_i$ is distributed according to some known continuous probability distribution that we can choose (Gaussian, ...
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Probability of the surface area of a pentagonal prism being greater than the volume
Consider a unit cube. Points $WXYZ$ are on the sides of the bottom face of the unit cube. Point $I$ is inside of the bottom face of the unit cube such that $WXYZI$ forms a pentagon. Point $M$ is ...
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What is the probability of an angle to be obtuse in a triangle (presumed to be on a plane rather than curved surface)?
Here's the progress I (think I) have made.
EQUILATERAL TRIANGLES
For Equilateral triangles, the probability is zero for any angle.
ISOSCELES TRIANGLES
For Isosceles triangles, when defining one of the ...
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Average Minimum Distance between Curve and Circle
I aim to formalize the average minimum distance between any point on a circle with a radius $r$ and an infinitely long curve.
I know the location of the curve and circle, but there is no formula for ...
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Conjecture on bounding probabilities
For an integer $m\geq 1$, consider the $m$th root of unity $\omega:=\exp(2\pi\mathrm i/m)$, and let $I\subseteq\{1,\dots,m\}$. Let $X_i$($i\in I$) be independent real random variables with mean $0$ ...
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Distribution of distances between random points on spheres
This is in relation to the following blog post by John Baez:
Random Points on a Sphere (Part 1)
It is claimed that, for two unit quaternions $x,y\in S^3$, uniformly randomly selected, the probability ...
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The Circle in a Circle problem-Probability that circle C2 passing through 3 randomly selected points in circle C1 lies completely inside C1. [duplicate]
Consider a circle. Three points are chosen at random inside the circle. What is the probability that the circle which passes through these three points while lie totally inside the original circle?
In ...
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Probability of forming a triangle within certain parameters
Stuck at problem 7.19, from "Understanding Probability by Henk Tijms". The problem statement is :
You choose a number $v$ at random from (0, 1) and next, a number $w$ at random from (0, $1 −...
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Volume marginals and surface marginals of a convex body
Given a convex body $K\subset \mathbb{R}^n$, the marginal distribution along an axis $l$ is the function that measures the intersection of $K$ with hyperplanes perpendicular to $l$.
My question is ...
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n-dimensional sphere, show that their are exponentially many points of distance > 1
I realy need help with part b). Part a) I could proof on my own but I'm really stuck at part b) or rather i have no idea how to start.
We denote the $n$-dimensional sphere by $S_n = {x ∈ R^{n+1} : ||x|...
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Expected Value of volume of any convex body excluding some points?
Consider the bounded region given by $A \in \mathbb{R}^n$. Let it be given that we have uniformly sampled $k$ i.i.d. points ${P_i} \sim U(A)$ where $k$ is some constant.
Now $S \subseteq A$ be any ...
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Probability that the sum of $n$ unit vectors (2D dimention) has length less than one
It is easy to show that the probability that the length of the sum of two random unit vector in a 2D-plane is less than one is $\frac13$.
As the picture above, assume the first vector is $\bar{AB}$ (...
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Viewing sides of a hexagon in a circle
I need help with this problem:
Take a circle with radius r, and place a regular hexagon of side length 2 so that the circle and hexagon are concentric. The probability of picking a point on the ...
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Find CDF of minimum dependent identical distributed random variables
I'm a post-graduate researcher in Telecommunications and am currently studying Geogeomatric stochastic's applications.
In the process of building systems, I faced the challenge of finding the minimum ...
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Probability of a point in a square
Suppose I have on the map a square of dimensions 100 x 100 meters of which I know its center (by means of its latitude and longitude, (L,l)). Let's imagine that I am given a point of coordinates (X,Y) ...
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Probabilistic vs Geometric Theory of Integration
Motivating Question: Let $X$,$Y$ be independent standard uniform random variables. How does one show, rigorously, that
$$
\mathbb{E}[X \mid X+Y = 1] = \frac{1}{2}?
$$
I would be interested in hearing ...
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a vague step in the proof of variance of geometric distribution
I've been reading a probability textbook recently and get stuck by the following steps in a proof of variance of geometric random variable, the idea is to use formula:$Var(X)=E[X^2]−E[X]^2$, $E[X]$ on ...
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Does the Mean of a Convex Body Shift when the Body Shifts?
Suppose you put a strictly positive (supported on all $\mathbb{R}^n$) probability measure $\psi$ on $\mathbb{R}^n$. Suppose its density has only one local maximum from which the density decreases in ...
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Confusion about definition of statistical model
Let $X$ be a sample space and define a probability distribution function $p:X\to \mathbb{R}$ such that $P(x)\geq 0$ and $\int p(x)dx=1$,let $S$ be a family of probability distributions on $X$ ,suppose ...
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How much must the casino charge for this game for a non-negative expected profit?
St. Pete's Casino is introducing a new dice game called "Guess and Win." After paying (not betting) a fixed ante of $x$ dollars, the player guesses what the next roll of a fair six-sided ...
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Projection of a Gaussian random vector onto the unit $\ell_1$ ball
Let $Z_n \in \mathbb{R}^n$, $Z_n \sim N(0, I_n)$ be a gaussian random vector, where $I_n$ is the identity matrix. The unit ball is defined as
$$ L_1 = \left[X \in \mathbb{R}^n: \| X \|_1 \leq 1 \...
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$L^1$ boundedness of scaled nearest neighbour distance for i.i.d. points in Euclidean space.
Suppose $X_1,\ldots,X_n$ are i.i.d. random vectors in $\mathbb{R}^d$ ($d \geq 2$) from some density $f$. I want to show that $\sup_{n \geq 2} \mathbb{E} n^{1/d}\min_{2 \leq j \leq n} ||X_j-X_1||_2 <...
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A Weighted Gaussian Inequality: $E[\frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n \sigma_i^2x_i^2} ] \ge \frac{\sigma_n^2}{\sum_{i=1}^n \sigma_i^2}$
Given $\sigma_1 \ge \dots \ge \sigma_n \ge 0$,
and independent random gaussian variables $x_1, \dots, x_n \sim \mathcal N(0,1)$,
I want to show:
$$
\mathbb E\left[
\frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n ...
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How to formalize this problem? Probability mass on a given manifold.
I am having a hard time figuring out how to formalize this problem, so I am asking for references that I can follow.
Let's say that $X$ is a random variable defined on the probability space $(\mathbb{...
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How to show the following collection of set is an Algebra
suppose $\Omega=\{0,1\}^\mathbb{N}$ is the outcome space of tossing an unbiased coin infinitely many times,
Define a map $\pi_n\colon\Omega\to\Omega_n$ denote the projection on the first n coordinates,...
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The Coin Problem (Probability)
Suppose there is a square room of side $d$ and we draw on the floor a circumference of radius $\frac{1}{5}d$ whose center concur with the center of the square room. If we throw a coin of radius $\frac{...
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Magnitude Of Spherical Simplex Centroid Is Decreasing
Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
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Equidistribuion of distances of integer points to a circle
I have noticed in the following graph that the distance
between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1$:=Circle with radius 7 and shell with thickness 1) and the nearest point on the inner ...
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Decomposition of a random vector into a linear sum of bounded random variables
Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{\bf{y}\in \mathbb{R}^n|\bf{y}=A\bf{x},\bf{x} \in [-1,1]^m\}$.
Consider a ...
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Buffon's needle with nonunifom distribution of drops
The famous Buffon's needle problem relates the area of a region $C$ (or length of a curve $C$, in 1D) to the expected number of times a randomly dropped needle will fall on the region.
This makes ...
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Determining a probability function for a sum of N i.i.d. geometric distributions between where N is discrete with a geometric pf [closed]
I was completing revision for an upcoming task and this question was presented. Was hoping for some insight!
The random variable $N$ is discrete, with probability function
$$f_N(n)=\begin{cases}
p(1-p)...
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Take the uniform distribution on a sphere, and project it to a plane in the Riemann sphere's way, what's the resulting distribution?
Title explains it 90%.
"Uniform distribution on a sphere" means the continuous distribution, not Fibonacci lattice.
There might be two possible interpretations of Riemann sphere (plane ...
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