Sylvester's Simplex Probability for Beta-Type Distributions
Exact formulas for a classical question in geometric probability: given d+2 independent random points in d-dimensional space, what is the probability that their convex hull is a simplex — equivalently, that one point falls inside the hull of the others? The answer is worked out for three point distributions, the multivariate Gaussian and the beta and beta-prime distributions, by reducing the probability to expected internal-angle sums of random simplices. In the Gaussian case the probability equals twice the sum of the solid angles of a regular (d+1)-dimensional simplex.
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 note, we compute the probability that their convex hull is a simplex focusing on three specific distributional sett…