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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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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37 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
44 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
42 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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1answer
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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1answer
41 views

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
156 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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3answers
1k views

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
86 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 >s ...
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0answers
25 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 ...
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1answer
93 views

Numbers $\alpha$ and $\beta$ are selected from interval $[0,1]$. What is the probability that $x^2+\alpha x + \beta ^2=0$ has real roots?

I know that discriminant must be greater than zero , so we have : $\alpha ^2-4\beta^2\geq 0$ $\alpha^2\geq4\beta^2$ $\alpha\geq 2\beta$ We draw a function $\alpha - 2\beta = 0 $ and our condition ...
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1answer
55 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
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2answers
94 views

Understanding the geometric distribution

Simple question that has to do with the interpretation of the geometric distribution and frequency function: $P (X=k) = (1-p)^{k-1}p $ for $k = 1,2,3... $ where we are interpreting X as being up to ...
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1answer
34 views

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent? Y will always depend on X . NO ? i know geometric ...
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1answer
69 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. ...
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1answer
120 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 ...
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1answer
209 views

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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40 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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46 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
38 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 - \prod_{n=1}^{\infty}{1-p(1-...
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3answers
299 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 $1cm$...
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39 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
177 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
66 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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2answers
41 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
69 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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186 views

Expected no. of 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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1answer
92 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
13 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
247 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
46 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
199 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
87 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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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.
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189 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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2answers
51 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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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 ...
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174 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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31 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
133 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 1....
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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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621 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: http://www.skytopia.com/...
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1answer
337 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
220 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
166 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
163 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
352 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 ...