All Questions
Tagged with geometric-probability random-variables
17
questions
3
votes
2
answers
99
views
Probability and average height of two lines intersecting above the x-axis
For context, I know at least basic calculus and what random variables are, but not much about doing calculations with random variables. I came up with a problem for myself where there are two points ...
- 103
1
vote
0
answers
97
views
Decomposition of a random vector into a linear sum of bounded random variables
Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{\bf{y}\in \mathbb{R}^n|\bf{y}=A\bf{x},\bf{x} \in [-1,1]^m\}$.
Consider a ...
- 11
0
votes
0
answers
84
views
Probability in a 2-dimensional random walk
A random walker moves in 2 dimension space. It starts from origin. $(x_0,y_0)=(0,0)$.
The random walker moves 4 directions with same probability. It means $(x_{n+1},y_{n+1})=(x_n+1,y_n)$ with ...
- 95
1
vote
0
answers
83
views
Coupling probability and random walk game (2)
There are 3 players and one dealer in a casino. The dealer chooses a player randomly($p_1=\frac{1}{3}$). The chosen player tosses a coin($p_2=\frac{1}{2}$).
If the coin lands head, the chosen player ...
- 95
6
votes
2
answers
307
views
The distribution of areas of a random triangle on the sphere - what are the second, third, etc. moments?
Suppose that we choose three points independently and uniformly at random on the surface of a unit sphere as the vertices of a triangle, and consider the area of this triangle. Call this random ...
- 17.7k
3
votes
1
answer
665
views
Is a uniform distribution on a sphere always a norm-scaled normal $\mathcal{N}(0,I_d)$ distribution?
Let $U \sim Unif(S^{d-1}).$ I was wondering if it's true that, and if yes, how could we prove that:
$U = \frac{Z}{\|Z\|}$ where $Z \sim \mathcal{N}(0, I_d), $ i.e. a uniform distribution on a sphere ...
- 2,364
3
votes
1
answer
809
views
If $U$ is uniformly distributed on $S^{d-1} \subset \mathbb{R}^d$, what's the distribution of its orthogonal projection onto any vector?
Let $U \in S^{d-1} \subset \mathbb{R}^d$ follow a uniform distribution on a sphere. Let $v \in \mathbb{R}^d.$ Then is the orthogonal projection $U^{T}v=\langle U,v \rangle$ uniformly distributed, and ...
- 2,364
3
votes
2
answers
463
views
Point picking in a 1x1 square: probability of line segments connecting 2 random interior points to catacorner vertices intersecting in the square.
Like many of the best problems I found this one on twitter.
"In a square with side length 1, two random points in the square are connected by segments to two opposite vertices. How likely is it that ...
- 6,268
5
votes
2
answers
161
views
Special case of Bertrand Paradox or just a mistake?
I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
- 73
3
votes
3
answers
849
views
Choosing 2 points on a line
Two points are selected randomly on a line of length $L$ so as to be on opposite sides of the midpoint of the line. Find the probability that the distance between them is greater than $L/3$.
I was ...
- 1,106
8
votes
1
answer
122
views
Expected area of a random $n$-gon
Choose $n$ points $\{z_1, \ldots, z_n\}$ from the unit circle $\partial \mathbb{D} = \{z \in \mathbb{C}: |z| = 1\}$ uniformly at random, and let $P_n$ be the convex hull of the $z_i$'s. Let $X_n = ...
- 701
0
votes
1
answer
37
views
Finding maximum likelihood estimates
Practise Question
To find the ML, am I right in thinking you take the log of the function then differentiate it with respect to p?
I am a bit confused as to what to do with the data given and am not ...
0
votes
2
answers
481
views
Generate m outcomes of a geometric random variable using the cdf of the random variable
As part of an assignment, I have been asked to write an Octave function that will generate $m$ pseudorandom outcomes of a geometric RV $X$ with parameter $p = 0.55$ in 2 separate ways:
Directly using ...
- 293
6
votes
2
answers
1k
views
Average shortest distance between a circle and a random point lying in it
What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it?
This question is just idle curiosity. Basically, it's the same as finding the difference ...
- 2,387
2
votes
1
answer
169
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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