Future distribution networks are undergoing radical changes, due to the high level of penetration of dispersed generation and information/communication technologies, evolving into the new concept of the Smart Grid. Dispersed generation systems, such as wind farms and photovoltaic power plants, require particular attention due to their incorporation of uncertain energy sources. Further and significant well-known uncertainties are introduced by the load demands. In this case, many new technical considerations must be addressed to take into account the impacts of these uncertainties on the planning and operation of distribution networks. This paper proposes novel Bayesian-based approaches to forecast the power production of wind and photovoltaic generators and phase load demands. These approaches are used in a probabilistic short-term steady-state analysis of a Smart Grid obtained by means of a probabilistic load flow performed using the Point Estimate Method. Numerical applications on a 30-busbar, low-voltage distribution test system with wind farms and photovoltaic power plants connected at different busbars are presented and discussed.