On the Distribution of the Sum of n Non-Identically Distributed Uniform   Random Variables
  The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulae for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulae some special cases which have appeared in the literature. 
David M. Bradley and Ramesh C. Gupta