The evolving complexity of distribution networks with higher levels of uncertainties is a new challenge faced by system operators. This paper introduces the use of Gaussian mixtures models as input variables in stochastic power flow studies and state estimation of distribution networks. These studies are relevant for the efficient exploitation of renewable energy sources and the secure operation of network assets. The proposed formulation is valid for both power flow and state estimation problems. The method uses a combination of the Gaussian components used to model the input variables in the weighted least square formulation. In order to reduce computational demands, this paper includes an efficient optimization algorithm to reduce the number of Gaussian combinations. The proposed method was tested in a 69-bus radial test system and the results were compared with Monte Carlo simulations.