  Let $f_n$ be a function assigning weight to each possible triangle whose
vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides.
Suppose ${\mathcal T}_n$ is a random triangulation, sampled uniformly out of
all possible triangulations of $P_n$. We study the sum of weights of triangles
in ${\mathcal T}_n$ and give a general formula for average and variance of this
random variable. In addition, we look at several interesting special cases of
$f_n$ in which we obtain explicit forms of generating functions for the sum of
the weights. For example, among other things, we give new proofs for already
known results such as the degree of a fixed vertex and the number of ears in
${\mathcal T}_n,$ as well as, provide new results on the number of "blue"
angles and refined information on the distribution of angles at a fixed vertex.
We note that our approach is systematic and can be applied to many other new
examples while generalizing the existing results.
