Sylvester's problem for beta-type distributions

Abstract: 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 $(1-|x|^2)^{\beta}$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+|x|^2)^{-\beta}$ (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.