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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distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
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18 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 ...
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19 views

A sort of modified geometric series

I was wondering if there are any hints on how to manage this series $$ \sum_{i=2}^\infty \prod_{j=1}^{i-1} (1-cj^{\beta-1})=(1-c)+(1-c)(1-c2^{\beta-1})+(1-c)(1-c2^{\beta-1})(1-c3^{\beta-1})... $$ with ...
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16 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 $\mathcal{L}_n(x)$ ...
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28 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 ...
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1answer
18 views

Probability of eventual success in independent trials (close form expression)

Consider a sequence of independent trials with success probability $p$. The formula for eventual success, i.e., there will be at least one success eventually, is $$ q = 1 - ...
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3answers
277 views

Infinite points on a paper?

I remember solving questions like this: On a paper with dimensions $30cm$ x $21cm$ if a rubber (erasers)* is dropped, what is the probability that it falls over a grey shaded region of dimensions ...
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1answer
29 views

How can I find the PDF of this function of normal variables? Or what is the distribution of distances between two random points on a unit sphere?

How can I find the probability density function of the random variable $D = \frac{\sqrt{\left(x-\sqrt{x^2+y^2+z^2}\right)^2+y^2+z^2}}{\sqrt{x^2+y^2+z^2}}$ If x, y, and z are all independently ...
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1answer
53 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
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1answer
54 views

Concluding from limiting behavior

I've recently seen the following question on the internet: If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around? ...
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26 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 ...
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1answer
52 views

Expected area of an internal triangle determined by a random point in a triangle

A point M is chosen at random (uniformly) inside an equilateral triangle ABC of area 1. How to find the expected area of the triangle ABM?
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1answer
48 views

Expected nof children “at least one boy and at least one girl, with boy older than girl”

A couple decides to keep having children until Cond1: they have at least one boy and at least one girl, Cond2: with boy older than girl and then stop. Assume they never have twins, that the ...
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41 views

How long does one remain in the square

This is a slightly known question: There is a square of unit side. Its centre is $O$. Two movable points $X$ and $Y$ are placed randomly in the square. Let $A$ be the midpoint of $OX$ and $B$ ...
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1answer
24 views

Issue with sum of probabilities of probability distribution function of a geometric random variable

Is it possible that the sum of probabilities of geometric distribution for "$k = 1,...,n$", where k is number of trials until the first success, is not equal to 1? I'm asking this, because I encounter ...
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1answer
9 views

Proof that 2 geometric random variable is NB

can someone write me the proof of 2 geometric variable are negative binomial ? $X\sim G(p)$ and $Y\sim G(p)$ how can i proof that $Z=X+Y \sim NB(2,p)$?
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1answer
60 views

Probability of Intersecting Two Random Segments in a Circle

I designed this problem and tried to solve it but didn't solve. Choose four points $A$, $B$, $C$ and $D$ from inside of a circle uniformly and independent. What is the probability that $AC$ ...
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1answer
24 views

PMF for K, the number of trails up to, but not including, the second success

I'm taking an MIT OCW course on Probability. Question: Al performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of three fair coins. ...
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1answer
110 views

Three points on sides of equilateral triangle

Let's choose three points on the sides of an equilateral triangle(one point on each side) and construct a triangle with these three points. what is the probability that area of this triangle be at ...
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2answers
69 views

Probability of segment lying in circle

Given a circle of radius R: $x^2+y^2\le R$, find probability of horizontal segment with length $\frac{R}{2}$ lie whole inside this circle. Position of segment's center has uniform distribution in ...
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0answers
32 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.
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1answer
46 views

Probability of receiving a correct packet of N bits

When a packet is transmitted on a communications link, the probability that a bit in packet is received in error is p. Assume that the packet has N bits. Suppose the packet length is random i.e. N is ...
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0answers
17 views

Is there a way of solving this probability question without using the survival characteristic of a geometric random variable?

I have the following problem, and the author presents the solution by using the survival characteristic of a geometric random variable: However, I am not very familiar with this method and was ...
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2answers
33 views

Sum of geometric and Poisson distribution

Suppose I have $X \sim \mathrm{Geom}(p)$ and $Y=\mathrm{Pois}(\lambda)$. I want to create $Z = X + Y$. (Note: the $X$ begins at $0$ rather than $1$.) Is this possible, please? Then I would calculate ...
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77 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 ...
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133 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 ...
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19 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 ...
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1answer
49 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 ...
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44 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 ...
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5answers
407 views

Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary

Background : I happen to love solving tough problems. Problem is, I simply cannot answer some! It happened again today, as I attempted to solve the questions in this site: ...
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27 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 ...
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1answer
155 views

Probability of intersection of line segments

A pair of points is selected at random inside a unit circle and a line segment is drawn joining the two. Another pair is selected and a second line segment is drawn. Find probability that the two ...
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1answer
88 views

Probability of average distance from origin of unit circle less than half

Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than half?
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1answer
72 views

PDF of distance from the center of a random point in the unit disk

I found in a certain website (also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: $P(d)=2d$, but no one is giving the way to derive ...
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1answer
86 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 ...
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4answers
240 views

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is uniform) independently of others We construct new triangle B from ...
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3answers
171 views

Probability that one of a set of four points lies within the triangle formed by the other three

Given four points, each randomly chosen with a uniform probability distribution in the interior of a (WLOG unit) circle, what is the probability that (any) one of the points lies within the triangle ...
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1answer
28 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
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2answers
81 views

Probability of two points being a certain distance apart on a circle

Is the probability of two points being a certain distance $k$ apart on a circle of length $m$, with $0\le k<\frac{1}{2}m $, always the same for any $k$?
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1answer
137 views

Selecting a Random Point Inside a Cube

A point $P$ is selected at random inside a cube. Find the probability that $\angle APB \geq 135^o$, where $\overline{AB}$ is a body diagonal of the cube. I am not able to come up with the right ...
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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}} ...
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0answers
66 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 ...
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2answers
47 views

2 pairs of points on a circle.

Two pairs of points are randomly chosen on a circle. Find the probability that the line joining the two points in one pair intersects that in the other pair. I've been thinking over this problem, ...
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3answers
129 views

Expected area of a random triangle with fixed perimeter

I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle ...
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1answer
133 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 ...
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1answer
38 views

Bertrand paradox Random midpoint

http://en.wikipedia.org/wiki/Bertrand_paradox_(probability) The link above explains Bertrand paradox in probability. In "Random Midpoint method" Bertrand uses a concept that all chords whose ...
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1answer
71 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 ...
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1answer
122 views

Circular distribution of circles

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of ...
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1answer
72 views

expected size of a special set of random points in the unit square

Today I came up with this fun problem, but I'm having a hard time to solve it completely myself. The question is the following: Let's generate n random points ...
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1answer
51 views

probability for two vectors to lie on different regions created by hypeplane

Suppose we have two vectors $v_i,v_j$ and there is one hyperplane whose normal is chosen uniformly from the unit sphere. Then what will be the probability that $v_i$ lies on one side and $v_j$ lies on ...