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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Joint Distribution Chapter of P exam book—Discrete case. Problem 41.7 (p exam book by M. Finan)

Part of the question's solution was already posted here:$X$ Poisson distribution, $Y$ geometric distribution - how to find $P(Y>X)$? Michal's answer was $ P(X=n)=e^{−λ}*λ^n/(n!)P(Y>=n+1)$ $=(...
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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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Probability that the convex hull of random points is a triangle

Question: Given a fixed number $k > 3$ of random points in the plane, distributed according to a 2D standard normal distribution, what is the probability that all of them lie within the same ...
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Probability with numbers in [0,1] and sum of their squares

We choose two points in $[0,1]$. (A) Calculate the probability the sum of their squares is less than $1$. (B) Calculate the probability the sum of their squares is equal to $1$. Let $x,y \in [0,1]...
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5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : ...
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Proving that a positive-integer valued random variable has the lack of memory property iff it has a geometric distribution.

Suppose that $X$ is a positive-integer valued random variable with the lack of memory property which states: Given that $X>n$, then $\mathbb{P}(X=n+k) = \mathbb{P}(X=k)$. Consider the case ...
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Breaking one stick and balancing it on another

Take two sticks (not necessarily of the same length). Break one of them at a uniformly random point, support the other one at a uniformly random point, and place the pieces of the former on the ends ...
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Point inside three triangles.

Inside triangle E (the large red triangle) there are 2 smaller similar triangles. A ( the yellow triangle ) and B ( the blue triangle ), both of these smaller triangles have a base length that is 1 / ...
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Probability — segments

Inside a line segment $E$ with length $6$ unit , there are $2$ segments $A$ with length $2$ unit and $B$ with length $3$ unit . The position of $A$ is fixed with its left ...
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Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
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geometric probability — parallelograms

Inside a rhombus E with sides 10 unit and one interior angle less than 90 degree , there are 2 parallel ( with E ) parallelograms A and B , both can move freely and ...
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Expectation of overlapping triangles

Let E denotes a triangle PQR with PQ = QR = 2 unit and angle Q = 90 degree . L , M and N are mid-points of PQ , QR and RP respectively . Let A denotes triangle PLN , B denotes triangle LQM , C denotes ...
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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 ...
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Is it possible to build a new probability space so that the product of two independent Gaussian r.v. still be Gaussian in the new space?

I know that if $X$ and $Y$ are two independent normal random variables defined on the same probability space ($\Omega$, $\cal{F}$,$\cal{P}$), the product may not be normal, but is it possible to ...
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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 ...
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What is the average maximum value of a set of random numbers? [duplicate]

Let $a_1, a_2, a_3, \ldots, a_{10}$ be ten randomly chosen real numbers in the interval [0,1]. Let $m$ be the maximal value out of these 10 numbers. What is the expected value of $m$? (i.e. If i ...
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Choose 3 points A, B and C in a circle O

I have to get p,q and r. p = the probability of triangle ABC is an acute-angled triangle q = the probability of triangle ABC is a right-angled triangle r = the probability of triangle ABC is an ...
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Geometric Probability Problem, Random Numbers $0$-$1+$ Triangles.

Randy presses RANDOM on his calculator twice to obtain two random numbers between $0$ and $1$. Let $p$ be the probability that these two numbers and $1$ form the sides of an obtuse triangle. Find $p$. ...
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Probability that for two random points in unit ball, one is closer to the center than to the other point

If I have two points $p_1, p_2$ uniformly randomly selected in the unit ball, how can I calculate the probability that one of them is closer to the center of the ball than the distance between the two ...
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Can the sum be simplified? $\sum_\limits{x=y}^{\infty} {x \choose y}\left(\frac{1}{3}\right)^{x+1}$

Let: $$f(y) = \sum_{x=y}^{\infty} {x \choose y} \left(\frac{1}{3}\right)^{x+1}$$ Can this be simplified somehow? Is it a standard probability distribution? I can only get as far as: $$f(y) = \frac{...
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Probability space for zebras and their number of stripes

On a trip to Africa the researcher Alison notices that zebras with an even amount of stripes have double the probability to be seen than zebras with an odd amount of stripes. Let $E_n$ denote the ...
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How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
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47 views

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 ...
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55 views

Number of vertices of a random convex polygon

Take $n>2$ random points, chosen independently with uniform probability on $[0,1]\times[0,1]$. What is the probability $P(n,k)$ that the convex hull of these points is a polygon with exactly $2<...
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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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33 views

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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2answers
79 views

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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79 views

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 ...
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Choosing a measure when dealing with geometric probabilities

(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 ...
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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 ...
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Average node degree k in a geometric random graph is $\frac{\Pi r^2n}{l^2}$

The paper "Small-Worlds: Strong Clustering in Wireless Networks" (http://arxiv.org/pdf/0706.1063.pdf) is indicating empirically that the average node degree $<k>$ is (or can be approximated by) $...
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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 ...
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Expected error of simplifying to a geometric distribution

While reading an answer related to solving a problem with a geometric distribution, the following question occurred to me. The answer gives two possibilities for replying the OP's question. In the ...
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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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Position error probability distribution when distance and angle error distributions are zero mean Gaussian

In one problem we are estimating the position of an object from the measurement of its distance $\mathbf{r}$ from a point as well as its angle $\mathbf{\theta}$ from the reference direction. The error ...
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The expected value of the $k$th order statistic of iid geometrically distributed rvs, and its asympotic expansion.

I have read the paper Combinatorics of geometrically distributed random variables: Left-to-right maxima. In the paper, the largest order statistic $X_{n:n}$ (i.e., $\max\{X_1,X_2,\ldots,X_n\}$) is ...
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Distribution of number of points in lune of random area (PPP)

I have been reading ElSawy et al's paper "Characterizing Random CSMA Wireless Networks: A Stochastic Geometry Approach" and am unsure about a seemingly straightforward equation that appears in the ...
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How likely is it that a random plane through the origin will intersect positive space?

In an n-dimensional hyperspace, how likely is it that a randomly chosen plane passing through the origin will intersect "all-positive co-ordinate space"? (By "all-positive co-ordinate space" I mean ...
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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 ...
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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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Basic Probability: choosing points at random on a circle

So, I know that the probability that three randomly choosen points on a circle will be on a semi-circle is 3/4 (as is discussed here: Probability the three points on a circle will be on the same semi-...
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The “pepperoni pizza problem”

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...
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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 ...