Sylvester’s problem for beta-type distributions

Consider d + 2 i.i.d. random points X 1 , … , X d + 2 in 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.