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A Geometric Derivation of the Irwin-Hall Distribution

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  • James E. Marengo

    ()

  • David L. Farnsworth

    ()

  • Lucas Stefanic

    ()

Abstract

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

Suggested Citation

  • James E. Marengo & David L. Farnsworth & Lucas Stefanic, 2017. "A Geometric Derivation of the Irwin-Hall Distribution," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-6, September.
  • Handle: RePEc:hin:jijmms:3571419
    DOI: 10.1155/2017/3571419
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    References listed on IDEAS

    as
    1. Hidetoshi Murakami, 2014. "A saddlepoint approximation to the distribution of the sum of independent non-identically uniform random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 68(4), pages 267-275, November.
    2. S. Sadooghi-Alvandi & A. Nematollahi & R. Habibi, 2009. "On the distribution of the sum of independent uniform random variables," Statistical Papers, Springer, vol. 50(1), pages 171-175, January.
    3. David Bradley & Ramesh Gupta, 2002. "On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(3), pages 689-700, September.
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    Cited by:

    1. Nick Costanzino & Michael Curran, 2018. "A Simple Traffic Light Approach to Backtesting Expected Shortfall," Risks, MDPI, Open Access Journal, vol. 6(1), pages 1-7, January.
    2. Santos, Maria João & Curcio, Eduardo & Mulati, Mauro Henrique & Amorim, Pedro & Miyazawa, Flávio Keidi, 2020. "A robust optimization approach for the vehicle routing problem with selective backhauls," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 136(C).

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