  We present an algorithmic approach to estimate the value distributions of
random variables of probabilistic loops whose statistical moments are
(partially) known. Based on these moments, we apply two statistical methods,
Maximum Entropy and Gram-Charlier series, to estimate the distributions of the
loop's random variables. We measure the accuracy of our distribution estimation
by comparing the resulting distributions using exact and estimated moments of
the probabilistic loop, and performing statistical tests. We evaluate our
method on several probabilistic loops with polynomial updates over random
variables drawing from common probability distributions, including examples
implementing financial and biological models. For this, we leverage symbolic
approaches to compute exact higher-order moments of loops as well as use
sampling-based techniques to estimate moments from loop executions. Our
experimental results provide practical evidence of the accuracy of our method
for estimating distributions of probabilistic loop outputs.
