# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Random walk on a cube

Start a random walk on a vertex of a cube, with equal probability going along the three edges that you can see (to another vertex). what is the expected number of steps to reach the opposite vertex ...
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### Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
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### 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 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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### Random walking and the expected value

I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me. Lets say we have a square, with vertex's 1234. I can randomly walk to each ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...