pr.probability - convex hull of k random points

The number of $k$-dimensional faces $f_k$ on a random polytope is well studied subject, and you are asking about the $k=0$ case. The distributions that have probably received the most attention are uniform distributions on convex bodies and the standard multivariate normal (Gaussian) distribution. As Gjergji mentioned, Bárány has some of the strongest results in this area. In particular Bárány and Vu proved central limit theorems for $f_k$.

This Bulletin survey article is a good place to start.

One amusing point worth noting: if you look at uniform distributions on convex bodies, the answer will change drastically depending on the underlying body. The convex hull of random points in a disk, for example, will have many more points than the convex hull of random points in a triangle.