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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3
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1answer
93 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 ...
-2
votes
1answer
111 views

A variant of the Birthday problem [closed]

Find the probability formula and formula approximated by exponential formula for a variant of the $Birthday Problem$ section below. There is a group of men included $k$ and another group of women ...
1
vote
0answers
50 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 ...
1
vote
1answer
45 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 ...
0
votes
0answers
68 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 ...
4
votes
3answers
186 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 ...
0
votes
0answers
50 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 ...
76
votes
4answers
6k views

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 ...
0
votes
1answer
17 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 ...
1
vote
2answers
34 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 $(1-p)^k$?
1
vote
0answers
23 views

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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
69 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 ...
0
votes
0answers
23 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 $M_{X_{2}}(t) = \frac{(1/4)e^{t}}{(1-(1-(1/4)t)e^{t}}$ ...
1
vote
0answers
19 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 ...
0
votes
0answers
34 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 ...
0
votes
1answer
31 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 ...
1
vote
0answers
16 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 ...
1
vote
0answers
33 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 ...
0
votes
1answer
33 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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
30 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 ...
-1
votes
1answer
38 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 ...
2
votes
2answers
92 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 > ...
43
votes
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 ...
5
votes
1answer
82 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 ...
0
votes
0answers
24 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 ...
2
votes
1answer
82 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 ...
0
votes
1answer
51 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 ...
0
votes
2answers
67 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 ...
1
vote
1answer
33 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 ...
2
votes
1answer
64 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. ...
1
vote
1answer
113 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 ...
1
vote
1answer
112 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 ...
0
votes
0answers
31 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 ...
0
votes
0answers
27 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 $\mathcal{L}_n(x)$ ...
0
votes
0answers
37 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 ...
1
vote
1answer
27 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 - ...
2
votes
3answers
295 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 ...
0
votes
1answer
37 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 ...
1
vote
1answer
124 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 ...
1
vote
1answer
62 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? ...
0
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0answers
33 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 ...
0
votes
1answer
61 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?
0
votes
1answer
136 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 ...
0
votes
1answer
64 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 ...
0
votes
1answer
11 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 $Z=X+Y \sim NB(2,p)$?
8
votes
1answer
164 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 $AC$ ...
5
votes
1answer
39 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. ...
2
votes
1answer
171 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 ...