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The environmental need to curb distribution network losses and utilize renewable energy sources has created new challenges in estimation. High fidelity estimates are required even in the presence of significant uncertainty. Herein, we develop a new analytical probabilistic load flow method that, unlike existing analytical methods, is not based on a Taylor series approximation of the power equations. The method is exact for a set of distributions that includes the multivariate normal distribution. The method implementation is made scalable by casting all formulas into the framework of the popular backward/forward algorithm. The advantages of this approach are illustrated on a radial IEEE 32-bus test system. Significant improvements are observed in the presence of large power uncertainties and near the network power limits. Uniformly better estimation of power losses is achieved.