Probabilites of random geometric objects having certain properties (enclosing the origin, having an acute angle, being convex, ...); expected counts, areas, ... of random geometric objects.

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1answer
39 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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0answers
7 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 we ...
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0answers
24 views

Understanding the difference of distributions

From the set {x : 0 ≤ x ≤ 1}, numbers are selected at random and independently and rounded to two decimal places. a) What is the probability that 0.35 is obtained for the first time on the 10th ...
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0answers
21 views

Geometric Probability Question Without Calculus [closed]

Three people go to the same place the same day. Each shows up at a random time within a 12 hour span. The first remains at the place for 2 hours, the second remains for 1 hour, and the third remains ...
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0answers
10 views

Probability of error in two dimensional signal space

The likelihood decision rule for two dimensional signal space is $r_2$ > $r_1$ for $H_1$ hypothesis and $r_2$ < $r_1$ for $H_0$ hypothesis. The range of $R=[r_1~r_2]$ is $-\infty<r_1<\infty$ ...
0
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1answer
148 views

Project Euler #453 confusion

So I decided to give a shot on the #453 project euler problem but there is something that confuses me with the numbers given. I decided to start by calculating the possible arrangements of 4 vertices ...
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1answer
22 views

expected value and negative binomial distribution

Repeatedly roll a fair die until the outcome 3 has occurred on the 4th roll. Let X be the number of times needed in order to achieve this goal. Find E(X). My Attempt : Pr(getting a 3 on the 4th ...
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1answer
40 views

Question on geometric distribution

Suppose that the probability for an applicant to get a job offer after an interview is 0.01. An applicant plans to keep trying out for more interviews until she gets offered. Assume outcomes of ...
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1answer
110 views

How do I find $P(X+Y = k)$ for a geometric distribution?

If $X$ and $Y$ are independent identically distributed random variables where $P(X=k) = P(Y=k) = pq^{k-1}$ where $q = 1-p$. How do you find $P(X+Y=k)$? Is it acceptable to say that $$P(X+Y=k) = ...
2
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1answer
72 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
95 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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1answer
30 views

Subtraction between two independent geometric random variables

Let X,Y be two independent and geometrically distributed variables with same parameter p, compute the PMF of X-Y. How about P(X=Y)?
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1answer
352 views

Probability generating function of geometric distribution

For a geometric distribution with $p_{x}(x)=p(1-p)^x, x=0,1,2,3,...$ I have been asked to find the probability generating function. I know that the way to find this is by finding $E(s^x)$ (the ...
2
votes
1answer
467 views

finding probability generating function and the sum of two independent random variables

Let $X$ be a discrete random variable with probability mass function $$P_X(x) = p(1-p)^x,\qquad x=0,1,2,3,\ldots$$ (a) Find the probability generating function for $X$ and hence ...
0
votes
2answers
50 views

Showing that the Geometric distribution $E(X)=\frac 1p$

So I have $X \sim \text{Geom}(p)$ and the probability mass function is: $p(1-p)^{x-1}$ From the definition that: $\sum_{n=1}^\infty ns^{n-1}$ = $\frac {1}{(1-s)^2}$ How would I show that the ...
0
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1answer
791 views

Proving the lack of memory property of the Geometric distribution

Can someone help me prove this: For any positive integer $k$ and $x$: $P(X=k+x|X>x)=P(X=x)$
0
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1answer
57 views

Geometric distribution, showing that $P(X>x)=(1-p)^{x}$ from the pmf definition

This may be a silly question but I can't seem to solve this for the life of me! The question states: Let the random variable $X$ have a Geometric distribution with parameter $p$ and probability mass ...
0
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1answer
105 views

geometric series word problem help

Brennen has been playing a game where he can create towns and help his empire expand. each town he has allows him to create 1.15 times as many villagers. The game gave brennan five villagers to start ...
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0answers
69 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
0
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1answer
21 views

craps game odd with a pair of dice

in a dice game craps, Alex rolls a pair of fair dice. if he gets 7 on the first roll, he wins immediately. if the result is any number other than 7, he keeps rolling the dice until he gets that number ...
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0answers
74 views

Spatial distribution of bees

* Please please help! I still get stuck. We have a forest for bees, consisting of $4$ non-overlapping regions. $80\%$ of the bees seek honey in the forest while $20\%$ of the bees do so outside the ...
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1answer
136 views

Probability of finding a point on or in an $n$-dimensional unit sphere

If a point is chosen at random in an $N$-dimensional unit sphere, what is the probability of falling inside the sphere of radius $0.99999999$? What if $N=3$, $N=10^{23}$, or $N = \infty $? Okay, that ...
4
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2answers
87 views

finding the number of circles we get when randomly placing given patterns into a grid of squares

We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into ...
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1answer
281 views

Simple geometric distribution (solution verification)

The question is: In a hockey competition, a player scores $80\%$ of his shots. What is the probability that the player will not miss until his $10^{th}$ try? So I did the following ...
3
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1answer
140 views

Truchet tiles on a flattened cube

We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
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0answers
20 views

finding the number of square we get when randomly put patterns into a given table [duplicate]

the image of the three tile patterns is here. [http://imageshack.us/photo/my-images/211/solpd.jpg/]
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2answers
79 views

Geometric probability question

So there are 2 parallel lines 20 feet apart. A piece of pipe 20 feet long falls between the lines and one end is exactly 10 feet from one line. What is the probability that the pipe lies entirely ...
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1answer
64 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
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2answers
189 views

A person is selected at random from a population that has the following characteristics

A person is selected at random from a population that has the following characteristics: ...
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1answer
575 views

Expected area of the intersection of two and three circles

We pick randomly two points, $p_1(x_1,y_1)$ and $p_2(x_2,y_2)$ inside a circle of origin $S$ with radius $R$ and we draw two circle $C_{1-2} (p_1,\sqrt {|x_1-x_2|²+|y_1-y_2|²})$ and $C_{2-1} ...
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1answer
123 views

What is the expected size of the convex hull of $n$-points selected randomly in a $2d$-circle?

We know $n>2$ and worst case its a triangle, best case the points all lie on a circle. Can we generalize to higher dimensions? What's the probability that the size of the convex hull of $n$ points ...
4
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0answers
106 views

Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
7
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1answer
113 views

A lawn, a flower, a pipe and the neighbours

You have a square lawn and a precious flower in the centre. You want to make sure you water the flower, and you don't particularly care how much of the lawn you water. To please your aleatory ...
8
votes
1answer
511 views

Expected area of the intersection of two circles

If we pick randomly two points inside a circle of radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the ...
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5answers
3k views

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
2
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1answer
155 views

Expected area of the intersection of of triangles made up random points inside a circle, all the triangles must contain the origin

How to find the expected area of the intersection of a set of triangles made up $N$ random points that are picked uniformly inside a circle? The triangles must contain the origin of the circle. If ...
2
votes
2answers
492 views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
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2answers
228 views

independent random variables geometric distributon

Suppose that $X$ and $Y$ are independent random variables with the same geometric distribution, $\mathbb{P}(X=k)=\mathbb{P}(Y=k)=pq^{k-1}$ for $k\geq 1$, $q=1-p$. Find $\mathbb{P}(X=k \mid ...
4
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1answer
104 views

How is the number of points in the convex hull of five random points distributed?

This is about another result that follows from the results on Sylvester's four-point problem and its generalizations; it's perhaps slightly less obvious than the other one I posted. Given a ...
2
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1answer
129 views

What's the probability for two points to lie on the same side of the line joining two other points?

While trying to answer this question I realized that the probability for two points to lie on the same side of the line joining two other points is directly related to the probability for four points ...
0
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1answer
49 views

Optimal $p$ for biased coin?

You are given a biased coin with probability $p$ of getting $H$ and $1-p$ of getting tail. Each flip is independent of another. We keep flipping the coin until we get $4$ consecutive tails. For each ...
10
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1answer
258 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 ...
14
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1answer
264 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
9
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3answers
2k views

What is the probability that the center of the circle is contained within the triangle?

Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?
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2answers
341 views

Probability of circle given by randomly chosen diameter falling inside a square

Two dots are thrown into a square with side length 1 cm. The line ending in these two dots is the diameter of a circle. What is the probability that the circle lies in the square?
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1answer
510 views

Proof of 2nd Derivative of a Sum of a Geometric Series

I am trying to prove how $$g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}$$ or $\sum ak(k-1)r^(k-1) = 2a(1-r)^{-3}$. I don't know what I am doing ...
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2answers
311 views

Geometry Probability Question

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Hi everyone I found this interesting question; help is ...
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1answer
76 views

Ring thrown in space

A ring is thrown randomly in the space and let $A(t)$ be its position at moment $t$. Let us say that the moment $t_{0}$ is "twisted", if the ring $A(t_{0})$ is linked with the rings $A(t)$ for $t$ ...
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2answers
504 views

Binomial/Geometric Distribution explanation

I've found the following exercise in my Stats coursework. I only have solutions to it, but no explanation. And I would really like to know how to get to the answer. An urn holds 5 white and 3 black ...
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2answers
382 views

probabilty of random points on perimeter containing center

related question: probablity of random pick up three points inside a regular triangle which form a triangle and contain the center What is the probability that a (possibly degenerate) triangle made ...