In this chapter, we study properties of beta-type distributions on affine Grassmannians as introduced in Chap. 4 . We derive two types of results. First, we prove stability of beta-type distributions on affine Grassmannians under taking affine hulls. More precisely, we show that if \(m\geq 2\) random affine subspaces in \(\mathbb {R}^d\) are independent and have beta-type distributions from the same family, then their affine hull also has a beta-type distribution of the same kind. Second, we investigate densities of restrictions. That is, we study the intersection of a random affine subspace following a beta-type distribution with a fixed linear subspace (or, which is equivalent, with a uniformly distributed linear subspace). The intersection is again a random affine subspace, whose distribution we characterize.