Consider the setting of \emph{randomly weighted graphs}, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, properties of weighted graphs typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some properties albeit the problem of computing them in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the \emph{diameter} of a given weighted graph, yet, computing the \emph{expected} diameter of a given randomly weighted graph is \SharpP{}-hard even if the edge weights are identically distributed. In this paper, we define a family of properties of weighted graphs and show that for each property in this family, the problem of computing the \emph{$k^{\text{th}}$ moment} (and in particular, the expected value) of the corresponding random variable in a given randomly weighted graph $G$ admits a \emph{fully polynomial time randomized approximation scheme (FPRAS)} for every fixed $k$. This family includes fundamental properties of weighted graphs such as the diameter of $G$, the \emph{radius} of $G$ (with respect to any designated vertex) and the weight of a \emph{minimum spanning tree} of $G$.