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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14
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1answer
396 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
7
votes
1answer
162 views

Expected time to completely cover a square with randomly placed smaller squares

Suppose I have the unit square $[0,1]^2$ and I choose a point $(x_1, y_1)$ randomly in a uniform manner inside $[0,1]^2$ and draw a filled in square of side length $1/N$ with center $(x_1, y_1)$. And ...
3
votes
1answer
130 views

Expected length of minimum chord

You are given a circle of radius $1$. Suppose you pick $n$ independent points randomly on the circle and join neighboring points with lines to create chords. What is the expected length of the ...
2
votes
1answer
64 views

Probability via Geometry, applications and examples

People, I am going to teach a lesson including something about Probability via Geometry and, if you have, I would like to know some references or materials (or even some good ideas) that can help me. ...
2
votes
1answer
78 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
2
votes
1answer
132 views

Convexity of a region on probability simplex

Exercise 2.15 g of Boyd et al Convex Optimization book : On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1,p_2,p_3,\ldots,p_n)$ corresponds to a distribution for random variable ...
1
vote
1answer
113 views

Circuit probability question regarding sum of a random number of independent random variables

Suppose we have n circuits that operate in a home. Each one will live according to an exponential random variable with rate λ. If X denotes the time at which a circuit first dies (i.e. the first circuit ...
1
vote
1answer
93 views

Probability of a coin falling on the edges of a square

Let a coin be randomly (and uniformly) dropped onto a square on the floor. Assume the edge length of the square to be $ d $ and the radius of the coin to be $ r < d/4$. I know that the probability ...
1
vote
1answer
206 views

Geometric Probability- Circle and two points

A point $P$ is chosen $0.5$ units away from the centre of a circle of diameter $2$. Now two points are chosen randomly on the circumference of the circle. What is the probability that the triangle ...
0
votes
1answer
17 views

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 ...
0
votes
1answer
85 views

Geometric probabilities with rectangle

One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than ...
6
votes
0answers
206 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
4
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0answers
163 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
3
votes
0answers
51 views

Random convex shapes containing a ball

I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space. Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
2
votes
0answers
163 views

The Curvy Rebound: One of the most interesting (Geometric) Probability problems.

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also ...
2
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0answers
96 views

Packing a larger sphere with smaller spheres in high dimensions

We were discussing today the probability of leaving a point uncovered while trying to fill a larger sphere by randomly throwing in smaller spheres. Here's the argument: We are working in ...
2
votes
0answers
107 views

Expectation number of random points exactly on their convex hull

Suppose there are n random points uniformly distributed in a square, what's the expectation of the number of the points located exactly on the edge (or being vertexes) of their convex hull? What if ...
1
vote
0answers
52 views

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 ...
1
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0answers
23 views

What is the probability that a rod can be cut with the length of cut being 5 units?

A rod of length 10 units and breadth 3 units is cut as shown in figure. Assuming that the longest cut can be from A to C and B to D. What is the probability that the cut made is of length 5 units. The ...
1
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0answers
20 views

Limits problem with convolution of identically distributed random variables X and Y

Schaums probability and statistics book gives this problem: Let X and Y be identically distributed independent random variables with density function: f(t) = 1 0 \ge t \le 1, 0 otherwise Find the ...
1
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0answers
16 views

probabilistically segmenting a rectangle

I am trying to find ways to segment a image randomly, but drawn from a probalistic distribution of pre-determined areas to be cut through. First thought was to pick random points and run Covex hull ...
1
vote
0answers
33 views

Probabilistic proof for sphere covering upper bound

I would like to show an upper bound for the number of $d$-dimensional spheres needed to cover some closed, bounded subset of $\mathbb{R}^d$, like a cube or another sphere. I could do this by placing ...
1
vote
0answers
36 views

Generate random numbers from appropriate distributions

Generate random numbers from appropriate distributions to find the area of the region enclosed by the curves y = sin (cos(x)), y = 0, x = pi/2 , and x = -pi/2 and report the area.
1
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0answers
80 views

Positivity of density for sum of dependent random variables

Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with ...
1
vote
0answers
50 views

Geometric probability with square

Jhon and Simon have common bank account which has $720$ dollars. Each of them has to buy a gift independently from other (Gift cost $< 360$ dol). What is the probability that after shopping there ...
1
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0answers
24 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
1
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0answers
50 views

Shortest path length when edge length is limited

$N$ nodes are uniformly distributed in a square whose side length is $1$. There exists an undirected edge between two nodes, if and only if the distance between them is less than or equal to $r$. Here ...
0
votes
0answers
12 views

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 ...
0
votes
0answers
68 views

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 ...
0
votes
0answers
50 views

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 ...
0
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0answers
23 views

Question About Moment Generating Functions

Question: Suppose that the random variables, $X_{1}$ and $X_{2}$ have the mgf's $M_{X_{1}}(t) = \frac{(1/2) e^{t}}{1-(1-(1/2))e^t}$, and $M_{X_{2}}(t) = \frac{(1/4)e^{t}}{(1-(1-(1/4)t)e^{t}}$ ...
0
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0answers
34 views

Geometric solution needed - Bus arrives randomly between 3pm and 330pm. Man goes randomly and only waits 5 minutes

This question asks for a geometric probability answer. A bus arrives randomly between 3pm and 330pm. A man decides he will go randomly to this location between these two times and will wait at most 5 ...
0
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0answers
24 views

Binomial/Geometric/Bayes perspectives on coin tosses?

? So I have the following question which I am trying to figure out/verify answers. a) I used the binomial probability mass function with n= 10 and p = 0.5 to determine the values. I think a success ...
0
votes
0answers
31 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
0
votes
0answers
27 views

Inequalities for Laguerre polynomials

The following inequality holds, $$ \Big( 4\int_0^\infty rdr \big|\mathcal{L}_1(4r^2)\big|e^{-2r^2}\Big)^3 \geq 4\int_0^\infty rdr \big|\mathcal{L}_3(4r^2)\big|e^{-2r^2}, $$ where $\mathcal{L}_n(x)$ ...
0
votes
0answers
37 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
25 views

How is the Dehn invariant related to the mean width?

Reading Ravi Vakils Monthly article of february 2011 and watching the video; he mentions that the Dehn invariant is related to the linear invariant measure $\mu_1$ of geometric probability. The Klain ...
0
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0answers
31 views

The expected number of polygons created as a result of the intersection between randomly placed rectangles inside a square

How can I compute the expected number of polygons created as a result of the intersection of $k$ rectangles of area $B$ each, which fall randomly inside a square of area A? Regarding how randomness ...