Propagation of input uncertainties in particle transport and the distribution of the sum of n independent stochastic variables a generalization of the Irwin-Hall distribution

We show, under wide hypothesis, that the problem of forward propagation of uncertainties in a physical model reduces to a well known problem in probability theory: the determination of the weighted sum of N independent stochastic variables obeying arbitrary distributions supported on bounded intervals. We determine various exact expressions for the probability distribution of the sum, possibly weighted, of N independent stochastic variables obeying arbitrary polynomial distributions on different intervals. These exact results provide the mathematical basis for the treatment of the given problem. We discuss the relevance of these findings for parameter uncertainty quantification in the context of particle transport computations. As a byproduct an explicit formula for the convolution of an arbitrary number of polynomials supported on bounded intervals is obtained.