# Tagged Questions

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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### Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
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### Average distance between two points lying on boundary of a square

Calculate the average distance between two points lying on the boundary of a unit square. I tried to approach it in the same way that this video does, but I couldn't really wrap my head around how ...
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### Find $\mathbb{P}(X>Y)$ given the distribution

Random variable (X,Y) has a uniform distribution over a triangle with vertices at $(1,0),(0,1),(-1,0)$. Find $P(X>Y)$ obviously it is going to be a double integral the answer i have in my answer ...
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### Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
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### Have I found an error in Williams' “Martingales” exercises?

I want to solve Problem EG.2. from Probability with Martingales: Planet X is a ball with centre O. Three spaceships A, B and C land at random on its surface, their positions being independent and ...
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### Expectation of the fill distance of $N$ random points in $[0,1]^s$

Let $x_1,\ldots,x_N$ be uniformly distributed points in $[0,1]^s, s \in \mathbb{N}$. What can be said about $$\mathbb{E} \left(\sup_{x \in [0,1]^s} \min_{i \in \{1,..N\}} \|x - x_i\|_2 \right)$$ ...
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### Probability that the convex hull of random points is a triangle

Question: Consider a fixed number $k > 3$ of random points in the plane, each independently distributed according to a 2D standard normal distribution. What is the probability that the convex hull ...
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### What is the probability that the center of a odd sided regular polygon lies inside a triangle formed by the vertices of the polygon?

Three distinct vertices are choose at random from the vertices of a given regular polygon of $2n+1$ sides. Each of those three vertices will determine a triangle. What is the probability that the ...
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### Probability that n points on a circle are in one quadrant

Question Points $A$,$B$ and $C$ are randomly chosen from a circle, What is the probability that all the points are in one quadrant ($\frac{1}{4}$ circle)? My try Using this answer about ...
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### Cover a line segment randomly with smaller line segments [closed]

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
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### What is the probability of uniformly sampling a point in d-dimensional hypercube?

Let us consider a hyper-cube whose length is l units along each of its d-dimensional structure. It is desired to uniformly sample a point inside the hyper-cube. How to do uniform sampling and what ...
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### Average Perimeter With n Points on the Unit Circle

A couple days ago, a friend challenged me to solve a problem: You have N vertices, each randomly placed on the edge of a unit circle. What is the formula (given N) that yields the average perimeter ...
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### Probability of triangle to be obtuse

Two points $A,B$ fixed on a plane(distance = 2). C - random choosen point inside circle with radius $R$ with center at the center of $AB$ Find probability of triangle $ABC$ to be obtuse My ...
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### Probability of a point lying between two other points on a line segment

On a line segment AB, 3 points - X, Y and Z are chosen at random. What is the probability that Z lies between X and Y?
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### Directional Statistics: Computation and Interpretation of Histograms of Directional Data

Suppose $v$ is a unit random vector in $R^n$ and consider a set $V = \{v^1,v^2,\dots\}$ of samples of $v$ gained either through experiments or simulations. How do we compute and interpret the ...
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### Random walk in high dimensional space with stationarity

I have a vector of high dimension ( say 100). When I take a random walk ( i.e add a step value to each components of the vector, the step value being drawn randomly drawn from standard normal ...
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### Average area of a random circle inside a triangle

Pick a random point inside a triangle $(0,0)(1,0)(0,1)$ (with uniform distribution) and draw a largest circle around it, which fully lies inside the triangle. What is the expected value of the circle ...
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### Average area of a random triangle inscribed in a semicircle

Let's say we have a triangle lying inside a semicircle ($R=1$), two vertices on the diameter ($x= \pm 1,~~y=0$), while the third somewhere on the circle in the first quadrant. It's pretty basic stuff,...
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### Mod of a random variable

I had this problem where I wanted to generate random variables (discrete) in a way that certain numbers were more probably than others (basically geometric) but since I wanted to use this number as an ...
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### Probability of an independent event according to past events

A binary communication system is used to send one of two messages: (i) message A is sent with probability 2/3, and consists of an infinite sequence of zeroes, (ii) message B is sent with probability ...
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### Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
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### Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the ...
(This is duplicate of the same question on MathOverflow where it was not approved.) The concept of geometric probability assumes that for each (uncountable) set $A$ corresponding to a random event ...