IDEAS home Printed from
   My bibliography  Save this article

A Geometric Derivation of the Irwin-Hall Distribution


  • James E. Marengo


  • David L. Farnsworth


  • Lucas Stefanic



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

    Download full text from publisher

    File URL:
    Download Restriction: no

    File URL:
    Download Restriction: no

    References listed on IDEAS

    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.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    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).

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hin:jijmms:3571419. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Mohamed Abdelhakeem). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.