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Sylvester’s problem for beta-type distributions

Author

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  • Gusakova, Anna
  • Kabluchko, Zakhar

Abstract

Consider d+2 i.i.d. random points X1,…,Xd+2 in Rd. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: •[(i)] the distribution of X1 is multivariate standard normal.•[(ii)] the density of X1 is proportional to (1−‖x‖2)β on the unit ball (the beta distribution).•[(iii)] the density of X1 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.

Suggested Citation

  • Gusakova, Anna & Kabluchko, Zakhar, 2025. "Sylvester’s problem for beta-type distributions," Statistics & Probability Letters, Elsevier, vol. 226(C).
    • Handle: RePEc:eee:stapro:v:226:y:2025:i:c:s0167715225001270
    DOI: 10.1016/j.spl.2025.110482
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    References listed on IDEAS

    as
    1. Zakhar Kabluchko & Daniel Temesvari & Christoph Thäle, 2019. "Expected intrinsic volumes and facet numbers of random beta‐polytopes," Mathematische Nachrichten, Wiley Blackwell, vol. 292(1), pages 79-105, January.
    2. Ruben, Harold & Miles, R. E., 1980. "A canonical decomposition of the probability measure of sets of isotropic random points in n," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 1-18, March.
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