In the context of Monte Carlo (MC) simulation of particle transport the goal of Uncertainty Quantification (UQ) is to become able to predict how non statistical errors affect the physical outcomes: these errors derive mainly from uncertainties in the physics data and/or in the model they embed, but also from uncertainties in the description of the experimental configuration under examination. In the case of a single uncertainty a simple analytical relation exists among its the Probability Density Function (PDF) and the corresponding PDF for the output of the simulation: then a complete statistical analysis of the results of the simulation is always possible. The extension of this result to the multi-variate case is examined, when more than one of the physical input parameters are affected by uncertainties: a generalized analytical relation exists among input and output PDFs, but some more sophisticated mathematical tools are needed to handle such expression.