Questions tagged [geometric-probability]
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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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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 ... complex-analysis probability-distributions geometric-probability- 69
Is there a continuous-valued metric for the convexity of 3D complexes?
I wonder if there a continuous-valued metric for the convexity of 3D complexes, such as simplicial or polyhedral complexes. It seems that, despite the good properties of the Euler characteristic, it's ... combinatorics algebraic-topology computational-geometry geometric-probability- 1
Probability problem my AP statistics teacher can't solve
This is a challenge problem that my AP Stat teacher can't solve, so I am hoping that I can find an answer here. I am aware that you could use a computer to run simulations to get an approximate, but I ... probability geometric-probability- 33
Showing that the intrinsic mean of continuously distributed data on the unit circle is almost surely unique
The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 7. Below is a screenshot of the authors' proof that the intrinsic mean ... probability probability-distributions proof-explanation geometric-probability- 239
projection of a non-zero mean Gaussian vector into a Ball
Let $d$ denote the dimension, $\mathbf{B}_d$ denote the ball of radius one in $\mathbb{R}^d$. For $x\in \mathbb{R}^d$ let $\Pi_{\mathbf{B}_d}(x) = \frac{x}{\max\{1,\|x\|_2\}}$. Consider a fixed vector ... probability projection gaussian geometric-probability- 561
If two points are chosen at random on the circumference of the circle, find the probability that the selected points form the diameter of the circle.
Question: If two points are chosen at random on the circumference of the circle, find the probability that the selected points form the diameter of the circle. My thoughts: For the $2$ points to be ... geometric-probability- 1,539
Finding the probability of point S forming a triangle with A and B in a square.
Let $S$ be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$. Let $P$ be the probability that the segments $AS$, $SB$, and $AC$ are congruent to ... probability geometry geometric-probability- 1,006
On Chaining Probabilities and Expected Lengths in $\mathbb{R}^3$
[I have recently asked a similar question; this is a tidy-up of the now-deleted one] Assume that we have $p_1, p_2, \dots, p_n \in [x_1, x_2] \times [y_1, y_2] \times [z_1, z_2] \subset \mathbb{R}^3$ ... probability-theory probability-distributions geometric-probability- 25
Probability of a point sampled from a ball lying in a spherical cap/segment
What is the probability that a point sampled from an n-ball lies in a spherical segment in the ball? Alternatively, what is the probability that a point sampled from an n-ball lies in a spherical cap ... probability geometry volume monte-carlo geometric-probability- 121
Volume of a spherical segment in high dimensions
Consider 2 parallel hyperplanes of the type $0 \leq \langle w,x \rangle + b$ and $\langle w,x \rangle + b < c$ where $x, w \in \mathbb{R}^n ; b,c \in \mathbb{R}$ cutting an $n$-ball with radius $r$ ... geometry volume spheres monte-carlo geometric-probability- 121
Probability that the average of a binary sequence deviates from $\frac{1}{2}$.
Is there a known estimate in terms of $n\in\mathbb{N},\varepsilon>0$ of the probability that a random sequence $x_1,\dots,x_n$ with $x_i\in\{0,1\}$ satisfies that for any $k$ with $2k<n$ we have ... probability statistics geometric-probability- 2,241
Probability of a triangle inside a square
Question If we have the square with vertices at the $4$ corners of $(0,1)^2$, and we choose a random point $z$ inside the square, the triangle is between $(0,0)$, $(1,0)$ and $z$, what is the CDF and ... probability statistics probability-distributions geometric-probability- 77
Geometric probability. On the interval [0, 5] we randomly and independently choose two numbers that divide this interval into three sections
On the interval [0, 5] we randomly and independently choose two numbers that divide this interval into three sections. The smaller number is denoted by a and the larger number by b. What is the ... probability geometric-probability- 13
Is the induced Fisher information metric equal to the Fisher metric of a submanifold?
If $M = \{p_\theta : \theta \in \Theta \subset \mathbb R^d \}$ is a statistical manifold parametrized by $\theta$, with the Fisher information metric \begin{equation*} g_{ij}(\theta) = \int_{\mathcal ... statistics differential-geometry geometric-probability fisher-information information-geometry- 21
Probability of picking 2 numbers between 0 and 1 to be within 1/2 distance of each other?
Problem: What's the probability of picking 2 numbers, x & y, between 0 and 1 such that they will be within the distance of $\frac{1}{2}$ of each other? In other words, $\Pr(\text{distance between ... probability area geometric-probability- 178
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