Skip to main content
SpringerLink
Log in

An Analytical Expression for the Distribution of the Sum of Random Variables with a Mixed Uniform Density and Mass Function

  • Conference paper
  • First Online:
Models, Algorithms, and Technologies for Network Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 32))

Abstract

The distribution of the sum of independent random variables plays an important role in many problems of applied mathematics. In this chapter we concentrate on the case when random variables have a continuous distribution with a discontinuity (or a probability mass) at a certain point r. Such a distribution arises naturally in actuarial mathematics when a responsibility or a retention limit is applied to every claim payment. An analytical expression for the distribution of the sum of i.i.d. random variables, which have a uniform distribution with a discontinuity, is reported.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbit, C.J.: Actuarial Mathematics. The Society of Actuaries, Itasca (1986)

    MATH  Google Scholar 

  2. Bradley, D.M., Gupta, R.C.: On the distribution of the sum of n non-identically distributed uniform random variables. Ann. I. Stat. Math. 54(3), 689–700 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buonocore, A., Pirozzi, E., Caputo, L.: A note on the sum of uniform random variables. Stat. Probabil. Lett. 79, 2092–2097 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I, II. Wiley, New York (1957, 1968)

    Google Scholar 

  5. Haehling von Lanzenauer, C., Lundberg, W.N.: The n-fold convolution of a mixed density and mass function. ASTIN Bulletin 8(1), 91–103 (1974)

    Google Scholar 

  6. Hall, P.: The distribution of means for samples of size n drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 19(3/4), 240–245 (1927)

    Article  MATH  Google Scholar 

  7. Irwin, J.O.: On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 19(3/4), 225–239 (1927)

    Article  MATH  Google Scholar 

  8. Kremer, E.: Applied Risk Theory. Shaker, Aachen (1999)

    Google Scholar 

  9. Olds, E.G.: A note on the convolution of uniform distributions. Ann. Math. Stat. 23, 282–285 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  10. Potuschak, H., Muller, W.G.: More on the distribution of the sum of uniform random variables. Stat. Pap. 50, 177–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sadooghi-Alvandi, S.M., Nematollahi, A.R., Habibi, R.: On the distribution of the sum of independent uniform random variables. Stat. Pap. 50, 171–175 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are partially supported by LATNA Laboratory, NRU HSE, RF government grant, ag.11.G34.31.0057.

Author information

Authors and Affiliations

  1. Laboratory of Algorithms and Technologies for Network Analysis and Department of Applied Mathematics and Informatics, The National Research University Higher School of Economics, 136, Rodionova Street, Nizhny Novgorod, Russian Federation, 603093, Russia

    Mikhail Batsyn & Valery Kalyagin

Authors
  1. Mikhail Batsyn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valery Kalyagin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Batsyn .

Editor information

Editors and Affiliations

  1. Higher School of Economics, Dept. of Applied Math. & Informatics, National Research University, B. Pecherskaya 25/12, Nizhny Novgorod, 603155, Russia

    Boris Goldengorin

  2. Higher School of Economics, Department of Applied Mathematics, National Research University, Nizhny, Novgorod, 603155, Russia

    Valery A. Kalyagin

  3. , Department of Industrial and Systems Eng, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this paper

Cite this paper

Batsyn, M., Kalyagin, V. (2013). An Analytical Expression for the Distribution of the Sum of Random Variables with a Mixed Uniform Density and Mass Function. In: Goldengorin, B., Kalyagin, V., Pardalos, P. (eds) Models, Algorithms, and Technologies for Network Analysis. Springer Proceedings in Mathematics & Statistics, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5574-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation