Distribution and moments of the weighted sum of uniforms random variables, with applications in reducing monte carlo simulations

Abstract

We derive analytical expressions for the distribution function and the moments of the weighted sum where X_i are independent random variables with non-identical uniform distributions, for an arbitrary number of variables N and arbitrary coefficient values a_i These results are the generalizations of those for the regular sum of uniform random variables. Using the results, we examine the inadequacy of the central limit approximation for finite N We also discuss the savings in the cost of computing properties of the weighted sum using these results vs Monte Carlo simulations. We give an example of the application of the weighted sum to analyzing the effects of digitization error in computer vision.

Keywords:

• Central limit
• Digitization error
• Monte Carlo simulation
• Sum of random variables
• Uniform distribution