All Questions
Tagged with geometric-probability probability-distributions
103
questions
-2
votes
0
answers
28
views
Kolmogorov bound for comparison of Random Vector Projections on a Sphere [closed]
Let $n$ be a fixed integer and $X$ be a random vector in $\sqrt{n} S^{n-1}$ (the $\sqrt n$-radius sphere in $\mathbf{R}^{n}$) with a density $f$ which satisfies the following property:
$
\forall x \in ...
- 1
1
vote
0
answers
57
views
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
3
votes
1
answer
184
views
Is there a formal proof that points taken at random in a bound area are evenly distributed?
I am an amateur trying to understand how probability works on the euclidean plane.
Despite my efforts I couldn't find any formal proof that points taken at random in a bound area are evenly ...
- 31
3
votes
0
answers
69
views
How can I find the average distance between two points inside a torus
Suppose we have the torus with equation $1-\left(\sqrt{x^2+y^2}-3\right)^2=z^2$. We choose two randomly chosen points inside it with a uniform distribution. I want to find the average distance between ...
- 101
1
vote
1
answer
52
views
Average distance between random points inside a semisphere and a quarter-sphere
Suppose we have a semisphere of radius 1. We choose two random points inside it with a uniform distribution. That is, if we pick random points insed it, they will be uniformly distributed.
What is ...
- 101
4
votes
0
answers
118
views
Distribution of distances between two randomly selected points in a semicircle
Suppose we have a semicircle with radius $1$: We choose two random points with a uniform distribution, that is, that if we pick random points inside it, they will ...
- 101
0
votes
1
answer
34
views
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
1
vote
0
answers
43
views
Is there anyway to guarantee probability mass coverage?
If I have a probability density function on $\mathbb{R}^n$. I can sample $m$ points from it. Is there anyway to get an estimate of how much probability mass is covered by balls of radius $\delta$ ...
- 1,799
0
votes
1
answer
45
views
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
0
votes
0
answers
15
views
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 $0 < R_a < R_b$, ...
4
votes
1
answer
145
views
Random points in 2-sphere
Say we have the PDF of uniformly picking a random point in the unit $2$-sphere, with distance from origin of the point being $r$, given as
$$
f_R(r) = \begin{cases}
3r^2 & 0 \leq r \leq 1 \\
...
- 89
1
vote
4
answers
125
views
Calculate expected number of heads in 10+$\xi$ coin tosses (GRE problem)
This is a problem from a preparatory GRE preparatory GRE test made by guys form University of Chicago.
Problem:
A man flips $10$ coins. With $H$ the number of heads, and $T$ the number of tails, the ...
- 830
2
votes
1
answer
142
views
Singular values of uniform random points on hypersphere?
This question is motivated by a self-supervised learning problem in machine learning, but I'll try to strip out as many unnecessary details as possible. In this setting, we have large datasets and we ...
- 502