Sylvester's problem for beta-type distributions

Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ in $\mathbb R^d$. In this short note we compute the probability that their convex hull is a simplex in the following three cases: (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 (beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-\beta}$ (beta prime distribution). In the normal case, the above probability is twice the sum of solid angles of a regular $(d+1)$-dimensional simplex.
Comment: 12 pages