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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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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21 views

Determine p value of the statistics [closed]

Given X is a geometric random variable with pmf and cdf $$p(x) = 0.153(1-0.153)^{x-1} $$ and $$F(x) = 1-(1-0.153)^x$$ A statistics S is defined by $S = \min\{X_1, X_2,X_3,\ldots,X_n\}$ If the ...
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1answer
22 views

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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44 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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48 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 ...
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45 views

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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50 views

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 ...
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11 views

I-projection of a distribution P onto a family of distributions Q

How do I get the I-projection as obtained on page 276 of probabilistic graphical models principles and techniques by Koller and Friedman?
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2answers
22 views

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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20 views

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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30 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
76 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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1answer
74 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 ...
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6 views

Estimating P_Hats Geometric Distribution

I understand how geometric distribution in a classical or classroom setting. My case is quite different as it has n observations and m trials. So, how do I estimate p? Formula says - success/attempts ...
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1answer
26 views

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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2answers
23 views

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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108 views

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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1answer
117 views

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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1answer
31 views

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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3answers
67 views

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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1answer
20 views

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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2answers
45 views

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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1answer
26 views

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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1answer
40 views

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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21 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 ...
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1answer
110 views

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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1answer
72 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 ...
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1answer
47 views

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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73 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 ...
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3answers
212 views

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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1answer
61 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 ...
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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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1answer
19 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 ...
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2answers
35 views

Why the equation can be equal to $(1-p)^k$?

I was studying for geometric random variable, and I saw that $P(X>k)$ =$\displaystyle \sum_{i\ge k+1}p(1-p)^{i-1}$ =$(1-p)^k$ I don't understand why it can be equal to $(1-p)^k$?
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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 ...
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1answer
33 views

Find the mean of the Geometric distribution from the MGF

I am trying to show, using the MGF of $X$~$Geom(p)$, that he mean of this distribution is $\frac{q}{p}$ and that the variance is $\frac{q}{p^2}$. I know that the MGF of X is ...
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1answer
73 views

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 ...
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30 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 $M_{X_{2}}(t) = \frac{(1/4)e^{t}}{(1-(1-(1/4)t)e^{t}}$ ...
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22 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 ...
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36 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 ...
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1answer
33 views

geometric distribution after $n$ trials

I am supposed to find the probability of $x$ happening for the first time after $n = 1000$ trials with $p = 0.001$ So after $n$, I'm assuming that this means at most $n$? I know the generic way of ...
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0answers
19 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 ...
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0answers
42 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 ...
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1answer
34 views

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 ...
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1answer
39 views

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 ...
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31 views

Solve an expression involving a sigma

Suppose we have the following equation. $$\underset{x=1}{\overset{n}{\Sigma }} (0.4)(0.6)^{(x-1)} \geq 0.6$$ I seek to find a value for n. I can't show any attempts to solve this since I don't ...
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Find $E[E[X \mid Y]]$ where $X \sim Geo[Y]$ and $Y \sim Unif[0, \frac{n-1}{n}]$

We have the random variable $X$ distributed geometrically with parameter $Y$ which is itself a random variable uniformly distributed over the interval $[0, \frac{n-1}{n}]$. We want to find $E[E[X ...
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2answers
130 views

Lower bound for the cumulative distribution function on the left of the mean

Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. Let $a < \mu$ and consider the probability $$ F_X(a) = \mathbb{P}(X \leq a) = \mathbb{P}(X - \mu \leq a - \mu). $$ If $a > ...
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Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
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1answer
84 views

Show the following pdf is memoryless

I've been thinking about this around 2 weeks for the midterm. but still can't prove it. I used this $$ P(X > r+s | X > s) = P(X > r) = \mathrm e^{−\lambda r}$$ $$P(X > r + s) / P(x ...