All Questions
Tagged with geometric-probability probability-theory
99
questions
1
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0
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57
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Probability distribution for the perimeter of a random triangle in a circle
This is the same question as this question except that the random triangles do not need to necessarily touch the circle and I am only interested in the distribution for the perimeter. And I am ...
- 61
0
votes
1
answer
34
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Expected number and variance of number when sum exceeds 1 (Irwin-Hall distribution)
I am given a random variable (uniformly distributed) between 0 and 1. To this, I add a second such random variable. I keep on adding these variables until sum exceeds 1, and then stop. Let us call ...
- 342
0
votes
0
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18
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Measure transport by a random matrix
I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are ...
- 1,085
1
vote
0
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30
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Probability of two geometric conditions happening together
I have a problem combining geometry with probability, and I feel like I do not understand the basics to approach this problem.
Let there be two points $\mathbf{p}_1$ and $\mathbf{p}_2$ in $\mathbb{R}^...
- 95
0
votes
1
answer
45
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Probability Theory: Generating Functions of Random Variables
Let $X, Y$ be independent random variables with the geometric distribution with parameter
$p > 0$.
(a) Compute the mean of $Z = XY$.
I got that $E(Z) = 1/p^2$
(b) Compute the probability ...
0
votes
1
answer
159
views
Expected radius of throwing a dart at a dartboard
I am doing a problem that states: If you are throwing a dart at a circular board with radius $R$, what is the expected distance from the centre?
If $x$ is the expected radius, then it would be the ...
- 45
3
votes
2
answers
133
views
Multivariate Normal Distributions and the Uniform Distribution on the Sphere
Given a multivariate normal vector $X \sim N(0,I_d)$ (identity covariance matrix), it is well known that :$$\frac{X}{\|X\|_2}
$$
is uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb{...
- 3,915
1
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0
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112
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Expected area of inscribed triangle
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that formula: $S = \frac{1}{...
- 11
0
votes
0
answers
15
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How to define pdf of the distance to the point of the spherical cap?
Suppose we have a sphere centered at $(0, 0, 0)$, with the radius of $R_b$. We cut the sphere with the tangent plane centered at $(0, 0, R_a)$, where a dude is fixed on. (Here $0 < R_a < R_b$, ...
1
vote
0
answers
46
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Probability for the highest correlation between random vectors
Let $k_1,\dots,k_m\in\mathbb{R}^d$, denote by $\mathcal{D}:=\mathcal{N}\left(0,\frac{1}{d}I_d\right)$ and $[n]= \{ 1,\dots,n \} $. I am interested in the following probability:
$P_{ x_1,\dots,x_n\sim \...
- 993
11
votes
4
answers
1k
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A question of random points in a square and probability of intersection of their line segments
The following is a problem from PUMaC 2007:
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square
and draw the line segment from it to $(0,0)$. ...
- 403
3
votes
1
answer
67
views
Expected number of packs until two cards are collected
Say we want to collect two cards A and B, each appearing independently in the pack with probability $p$. Note that a pack may contain both A and B at the same time with probability $p^2$. I want to ...
- 33
0
votes
0
answers
51
views
Conditional probability, two vector valued RVs on a circle.
Given two vector valued RVs $ X,Y \in S_1$ (the unit circle) that are uniformly distributed. We choose a fixed vector $a \in S_1$, e.g. in x-direction: $a=e_x$.
We then sample once from both ...
- 1
0
votes
1
answer
109
views
Uniform Distribution of Chords
In the context of the Bertrand Paradox, I understand that different methods of choosing a chord lead to different probability density functions. For instance, selecting a chord based on a random ...
- 770
1
vote
1
answer
173
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Find CDF of minimum dependent identical distributed random variables
I'm a post-graduate researcher in Telecommunications and am currently studying Geogeomatric stochastic's applications.
In the process of building systems, I faced the challenge of finding the minimum ...
- 73