  Let $X_1,\ldots,X_n$ be i.i.d.\ random points in the $d$-dimensional
Euclidean space sampled according to one of the following probability
densities: $$ f_{d,\beta} (x) = \text{const} \cdot (1-\|x\|^2)^{\beta}, \quad
\|x\|\leq 1, \quad \text{(the beta case)} $$ and $$ \tilde f_{d,\beta} (x) =
\text{const} \cdot (1+\|x\|^2)^{-\beta}, \quad x\in\mathbb{R}^d, \quad
\text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes
and the expected number of facets of the convex hull of $X_1,\ldots,X_n$.
Asymptotic formulae where obtained previously by Affentranger [The convex hull
of random points with spherically symmetric distributions, 1991]. By studying
the limits of the beta case when $\beta\downarrow -1$, respectively $\beta
\uparrow +\infty$, we can also cover the models in which $X_1,\ldots,X_n$ are
uniformly distributed on the unit sphere or normally distributed, respectively.
We obtain similar results for the random polytopes defined as the convex hulls
of $\pm X_1,\ldots,\pm X_n$ and $0,X_1,\ldots,X_n$. One of the main tools used
in the proofs is the Blaschke-Petkantschin formula.
