All Questions
5
questions
5
votes
4
answers
439
views
Confusion on defining uniform distribution on hypersphere and its sampling problem
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ...
- 2,497
0
votes
0
answers
70
views
Integration by parts over $R^n$
Let $\mu$ be a probability measure over $\mathbb{R}^n$. Let $f$ and $g$ be two real-valued functions on $\mathbb{R}^n$. I would like to compute
$$\int_{\mathbb{R}^n} \nabla f(x) g(x) \, d\mu(x). $$
I ...
Quarth
2
votes
1
answer
101
views
Extending a probability measure on a manifold to the ambient space
Let $\pi$ be a probability measure defined on $(\mathcal{M}, \mathcal{B}(\mathcal{M}))$, where $\mathcal{M}$ is a smooth manifold and $\mathcal{B}(\mathcal{M})$ is the Borel sigma-algebra on it. Let $\...
- 5,247
1
vote
0
answers
18
views
Defining probability of having a certain normal in surface patch.
I'm neither an expert in probability theory or in differential geometry. But for a paper I've been reading I'm trying to formalize some concept by myself. This is idea I'm trying to formalize.
...
- 5,317
2
votes
1
answer
908
views
Uniformly distributed points on spherical surface
Let $x=(x_1,\ldots,x_n)$ be uniformly distributed on the $(n-1)$-dimensional spherical surface $S^{n-1}(n^\frac{1}{2})$ of radius $n^\frac{1}{2}$. I'm trying to show that as $n\to\infty$, $x_1$ ...
- 1,354