  Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and
$(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points
uniformly distributed in $T$ is the random convex chain $T_n$. A three-term
recursion for the probability generating function $G_n$ of the number
$f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal
polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in
$(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$,
$k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of
probabilistic consequences of this surprising and remarkable fact are discussed
in detail.
