On the moments of the volume for random convex chains

Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull $T_n$ of points $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain. In this paper we study the moments of the volume of random polytope $T_n$ and derive exact formulas for $k$-th moments for any integer $k\ge 0$. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of $T_n$, from which an alternative formula for the probability that $T_n$ has $k$ vertices follows.