  Probabilistic loops can be employed to implement and to model different
processes ranging from software to cyber-physical systems. One main challenge
is how to automatically estimate the distribution of the underlying continuous
random variables symbolically and without sampling. We develop an approach,
which we call K-series estimation, to approximate statically the joint and
marginal distributions of a vector of random variables updated in a
probabilistic non-nested loop with polynomial and non-polynomial assignments.
Our approach is a general estimation method for an unknown probability density
function with bounded support. It naturally complements algorithms for
automatic derivation of moments in probabilistic loops such
as~\cite{BartocciKS19,Moosbruggeretal2022}. Its only requirement is a finite
number of moments of the unknown density. We show that Gram-Charlier (GC)
series, a widely used estimation method, is a special case of K-series when the
normal probability density function is used as reference distribution. We
provide also a formulation suitable for estimating both univariate and
multivariate distributions. We demonstrate the feasibility of our approach
using multiple examples from the literature.
