Sylvester's problem for beta-type distributions

Consider $d+2$ i.i.d. random points $X_1,\dots ,X_{d+2}$ 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 $(1-\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{\beta }$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{-\beta }$ (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.