Sylvester's problem for beta-type distributions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909630865604608 |
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| author | Gusakova, Anna Kabluchko, Zakhar |
| author_facet | Gusakova, Anna Kabluchko, Zakhar |
| contents | Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ in $\mathbb R^d$. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: (i) the distribution of $X_1$ is multivariate standard normal; (ii) the density of $X_1$ is proportional to $(1-\|x\|^2)^β$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-β}$ (the beta prime distribution). In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular $(d+1)$-dimensional simplex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00671 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sylvester's problem for beta-type distributions Gusakova, Anna Kabluchko, Zakhar Probability Metric Geometry Primary: 60D05, 52A22, Secondary: 52A40 Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ in $\mathbb R^d$. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: (i) the distribution of $X_1$ is multivariate standard normal; (ii) the density of $X_1$ is proportional to $(1-\|x\|^2)^β$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-β}$ (the beta prime distribution). In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular $(d+1)$-dimensional simplex. |
| title | Sylvester's problem for beta-type distributions |
| topic | Probability Metric Geometry Primary: 60D05, 52A22, Secondary: 52A40 |
| url | https://arxiv.org/abs/2501.00671 |