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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, Dover, New York.
Borwein, D. and Borwein, J. M. (2001). Some remarkable properties of sinc and related integrals, The Ramanujan Journal, 5(1), 73–89.
Chu, J. T. (1957). Some uses of quasi-ranges, Ann. Math. Statist., 28, 173–180.
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York.
Johnson, N. L., Kotz, S. and Balakríshnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Wiley, New York.
Jordan, C. (1979). Calculus of Finite Differences, 3rd. ed., Chelsea, New York.
Körner, T. W. (1988). Fourier Analysis, Cambridge University Press, Cambridge.
Leon, F. C. (1961). The use of sample quasi-ranges in setting confidence intervals for the population standard deviation, J. Amer. Statist. Assoc., 56, 260–272.
Mitra, S. K. (1971). On the probability distribution of the sum of uniformly distributed random variables, SIAM J. Appl. Math., 20(2), 195–198.
Naus, I. (1966). A power comparison of two tests of non-random clustering, Technometrics, 8, 493–517.
Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung, Springer, Berlin.
Olds, E. G. (1952). A note on the convolution of uniform distributions, Ann. Math. Statist., 23, 282–285.
Rényi, A. (1970). Probability Theory, North-Holland, Amsterdam.
Roach, S. A. (1963). The frequency distribution of the sample mean where each member of the sample is drawn from a different rectangular distribution, Biometrika, 50, 508–513.
Schwatt, I. J. (1924). An Introduction to the Operations with Series, University of Pennsylvania Press, Philadelphia.
Tach, L. T. (1958). Tables for cumulative distribution function of a sum of independent random variables, Convair Aeronautics Report M, ZU–7–119-TN, San Diego.
Author information
Authors and Affiliations
About this article
Cite this article
Bradley, D.M., Gupta, R.C. On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables. Annals of the Institute of Statistical Mathematics 54, 689–700 (2002). https://doi.org/10.1023/A:1022483715767
Issue Date:
DOI: https://doi.org/10.1023/A:1022483715767