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.

Geometric probability is a tool to deal with the problem of infinite outcomes by measuring the number of outcomes geometrically, in terms of length, area, or volume. In basic probability, we usually encounter problems that are "discrete" (e.g. the outcome of a dice roll; see probability by outcomes for more). However, some of the most interesting problems involve "continuous" variables (e.g., the arrival time of your bus

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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 booklet is $\int_0^{0.5} \int_y^{1-y} 1 dxdy$. Why is…
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Average Area of Convex Hull of N points in Unit Hypercube

Suppose we randomly pick $N$ points inside the unit hypercube in $\mathbb{R}^n$, and form their convex hull. What is the expected value of the volume of the convex hull? For example, in the case $N=n=2$, we are asking about the average length of a…
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How to define a uniform probability distribution over a convex polytope / polyhedra and add them?

Let $P$ be a convex 3d polyhedra / 2d polytope constrained by a set of linear inequalities $Ax<= b$. 1.How to define a uniform probability distribution over a polytope/polyhedra? Let us say we have multiple polytopes/polyhedra with uniform…
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What is a random set of points in $R^2$?

Given a finite set of $n$ points $S$ in $R^2$, its convex hull, $cvx(S)$, can be obtained with the aid of many algorithms. To numerically compare these algorithms and study their complexity I need to start with an $S$. The question is what is a…
Maesumi
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Proof of Barbier's Theorem

In the probabilistic proof of Barbier's Theorem, I'm not sure why the expected number of line crossings of a continuous curve is the limit of the expected number of line crossings of piecewise linear functions approximating it. The proof of the…
xisao
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Do the lengths of all three segments necessarily have the same distribution?

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have the same distribution?
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Buffon needle problem , scenario $\ell>d$

suppose we have the classic problem of buffon's needle , let $\ell$ be the length of the needle and $d$ the distance between the parallel lines . I have solved the case which $\ell \leq d$ and i understand why $P(\text{needle cross the line})=…
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Good introductory book in geometric probability

I recently came across the proof of the Buffon theorem and I was fascinated by geometric probability. Could someone indicate me a good introductory book? Maybe with many exercises?
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Asymptotic size of a given dominating set in a random geometric graph

We consider a random geometric (undirected) graph $G=(V,E)$ ($n=|V|$): to each vertex $u \in V=\{0,\ldots,n-1\}$ a random point $P(u) \in [a;b]^2$ is associated. two vertices $u$ and $v$ are connected iff $|P(u)-P(v)|\le 1$. Let $N(u)=\{v\in V,…
md5
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avg # of maximum intersections for m 1-dimensional segments with length L in a range [0,t]?

I have a discrete range, let's say $[0,T]$. I also have $m$ segments of length $L\leq T$. A segment is $seg=(a, a+L)$, with $0 \leq a \leq (t-L)$. The total number of possible configurations of my segments is therefore $(T-L+1)^m$. In each of…
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Two points selected randomly on a line of length L, both independent uniform random variables.

Let $X$ and$Y$ be the two points such that $X$ ~ $U(0,\frac{L}{2})$ and $Y$ ~ $U(\frac{L}{2},L)$ What is the probability that the distance between $X$ and $Y$ is greater than $\frac{L}{3}$? I know that it is easier to calculate $1-P(Y-X…
Jabernet
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Probability that coordinate of a dot within a square less than random parameter Z

From square with vertices (0;0), (0;1), (1;1), (1;0) random dot was taken. It has coordinates (a;b). a and b are inside interval [0;1]. For random parameter z that is between [0;1] find probability that P(min(a,b) < 2z). When I solve this task, I…
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Geometric Probability in MoreThan 3 Dimensions

I know that geometric probability works well when there are 2 or 3 variables involved. However, I am not sure how to use this method when there are more than 3 variables. For example: *Five friends decide go meet at a restaurant at a random time…
Jed
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Probability that distance of two random points within a sphere is less than a constant

Two points are chosen at random within a sphere of radius $r$. How to calculate the probability that the distance of these two points is $< d$? My first approach was to divide the volume of a sphere with radius d by a sphere with radius r. But it…
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Buffon noodle problem gives theoretical issues

Let $\Gamma$ be a rectifiable curve in plane, having length $l$. Denote by $X_{\Gamma}$ the random variable that represents the number of crossings between $\Gamma$ and a grid of $d$-spaced parallel lines, where $d$ is a positive real number. We can…
Bogdan
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