Sylvester’s problem for beta-type distributions

Anna Gusakova and Zakhar Kabluchko

Statistics & Probability Letters, 2025, vol. 226, issue C

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.

Keywords: Sylvester problem; Beta distribution; Beta prime distribution; Normal distribution; Random polytope; Random simplex; Regular simplex; Internal angle (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spl.2025.110482

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