ON THE DISTRIBUTION OF THE SUM OF n NON-IDENTICALLY DISTRIBUTED UNIFORM RANDOM VARIABLES

DAVID M. BRADLEY AND RAMESH C. GUPTA

Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, U.S.A., e-mail: dbradley@e-math.ams.org; rcgupta@maine.maine.edu

(Received August 28, 2000)

Abstract. 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.

Key words and phrases: Uniform distribution, probability density, convolution, Fourier transform, sine integrals.