The stochastic nature of chemical reactions involving randomly fluctuating
population sizes has lead to a growing research interest in discrete-state
stochastic models and their analysis. A widely-used approach is the description
of the temporal evolution of the system in terms of a chemical master equation
(CME). In this paper we study two approaches for approximating the underlying
probability distributions of the CME. The first approach is based on an
integration of the statistical moments and the reconstruction of the
distribution based on the maximum entropy principle. The second approach relies
on an analytical approximation of the probability distribution of the CME using
the system size expansion, considering higher-order terms than the linear noise
approximation. We consider gene expression networks with unimodal and
multimodal protein distributions to compare the accuracy of the two approaches.
We find that both methods provide accurate approximations to the distributions
of the CME while having different benefits and limitations in applications.