All Questions
Tagged with geometric-probability measure-theory
10
questions
1
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0
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41
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How to show the following collection of set is an Algebra
suppose $\Omega=\{0,1\}^\mathbb{N}$ is the outcome space of tossing an unbiased coin infinitely many times,
Define a map $\pi_n\colon\Omega\to\Omega_n$ denote the projection on the first n coordinates,...
- 181
1
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97
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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
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1
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28
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Mixture / Union of Measures defined on disjoint subsets
Suppose I have a partition of $\mathbb{R}^N$
$$
\mathbb{R}^N = \bigcup_{i \in \mathbb{R}} \mathcal{X}_i \qquad \qquad \text{with } \mathcal{X_i} \cap \mathcal{X}_j = \emptyset \text{ if } i \neq j
$$
...
- 5,247
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38
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Probability of the ratio of two sequences of iid normal random variables whose means go to infinity
Let $X_m, Y_m $ be iid $\mathcal{N}(m, 1)$ random variables. Let $m \to \infty.$
I feel the following are true: for every fixed $\epsilon > 0$
$$lim _{m \to \infty} P[1 < \frac{X_m}{Y_m} < 1 ...
- 347
2
votes
0
answers
74
views
Finite set of vectors approximating a unit ball.
I am having difficulty proving that a unit ball can be approximated with a set of finite vectors. Specifically, I want to bound the error of the following approximation.
Let $D$ be a uniform ...
- 57
2
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46
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Bounds on the minimum distance between a set of points
If I have a disk domain $\mathcal{D} \subset \mathbb{R}^{2}$ of radius $r>1$, and two sets of points $a_1,\dots,a_n \in \mathcal{D}$ and $b_1,\dots,b_m \in \mathcal{D}$ distributed uniformly at ...
- 2,797
1
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2
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104
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About setwise convergence on open sets of nearest neighbor algorithm.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $(\mathcal{X},d)$ be a complete, separable, locally compact metric space. Suppose that $X,X_1,X_2,X_3,... : \Omega\to\mathcal{X}$ are $\...
- 5,783
0
votes
2
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164
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A geometric probability question
Find the probability of distance of two points ,which are selected in $[0,a]$ closed interval, is less than $ka$ $k \lt 1$
What did I write :
$P(A)$ = (Area measure of set $A$)/(Area measure of set $...
- 1,313
1
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0
answers
110
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Positivity of density for sum of dependent random variables
Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with ...
- 817
1
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2
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356
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Integral of convex set
Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely,
why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
- 259