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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Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
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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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32 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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1answer
60 views

What is the probability that a random line meet both of the opposite side of a square? [closed]

If it is known that a random line meets a side of a given square, show that the probability that it also meets the opposite side is p = sqrt2— 1.
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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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25 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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75 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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95 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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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
28 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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32 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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353 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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21 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
66 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
64 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
50 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
68 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
210 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
124 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
24 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
62 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
116 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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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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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
43 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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122 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
112 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
33 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
67 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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40 views

Radon transform, Buffon's needle and Integral geometry

In all the literature that I have seen it is mentioned that these two are "branches" of integral geometry, but no where I can see the exact connection since one is connected with probability and the ...
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1answer
112 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
61 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
48 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 ...
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1answer
50 views

fixed length random chord outside of circle.

consider a uniform distribution on a unit circle, I construct a cord by the following steps: pick one endpoint A within the unit circle uniformly. points that are $0<d<1$ distance away from ...
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1answer
68 views

Probability that a triangle has an angle greater than 120 degrees

We've got a circle and we draw $3$ points, which form a triangle. Question: what is the probability that its greatest angle has more than $120$ degrees? Well, I have no idea how to do it. I know some ...
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31 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 ...
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1answer
35 views

Density on the square, expected value

Let $f: [0,1]^2 \rightarrow \Bbb R^{+}$ a density function on the square. I suppose that the random variable $X=(X_1,X_2)$ has the density f with respect to the lebesgue measure. I denote ...
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1answer
81 views

Random embedding of $K_4$ in the unit square

Suppose I embed $K_4$ (the complete graph on 4 vertices) randomly in the unit square (using the uniform distribution for the positioning of the vertices). $K_4$ is planar, but not any embedding of it ...
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1answer
131 views

A point is selected uniformly at random in the interior of a unit square…

From It, the altitude to each side of the square is drawn. For each side, a stick of the altitude's length is obtained. Determine the probability that you can select three of the sticks and arrange ...
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3answers
415 views

find a chance that all N points lie on the half circle. [duplicate]

We are given a circle with N randomly allocated points on it. Task is to find a chance that all N points lie on the one half of circle. I have drafted some solution: 1. Since there are no way to put ...
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1answer
893 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
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1answer
123 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
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1answer
154 views

Random Triangle Inscribed in a Circular Sector

Lately, I have been thinking about expected area and perimeter of a triangle inscribed in a 'partial' circle or circular sector with radius $r$ and truth be told, I couldn't answer these questions. I ...
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2answers
135 views

random circle with radius r on cartesian plane, probability of it not cutting x and y axis with intercepts.

I have a tough question here. Choose a circular disk of radius r on the cartesian plane. What's the probability it is not cut by horizontal lines with integer y intercept, or vertical lines with ...
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1answer
99 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 ...
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2answers
496 views

What is the average length of 2 points on a circle, with generalizations

I have earlier seen the question about finding the average length of two points and $n$ points inside the unit disk. But what about the more simple question, what happens if the points lie exactly on ...
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1answer
142 views

Expected value of maximum distance between points

Consider a two dimensional square domain ($S$) of size $l \times l$. We generate a point $\mathbf{x}_i = (x_i,y_i)$ in S with uniform distribution, i.e., the point is equally likely to be anywhere ...
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1answer
170 views

Three random points on a circle, CDF for second largest angle

Three points $A$, $B$, $C$ are chosen randomly on a circle. Let us consider angles $\alpha$, $\beta$, $\gamma \in [0, 2\pi)$ formed by consecutive pairs of points. Angles are reordered from the ...
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140 views

Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
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155 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 ...