Skip to main content
SpringerLink
Log in
Book cover

Applications of Interval Computations pp 227–244Cite as

Automatically Verified Arithmetic on Probability Distributions and Intervals

  • Chapter

Part of the book series: Applied Optimization ((APOP,volume 3))

Abstract

In this chapter we address two related problems:

  1. 1.

    Representing and operating on operands which are probability distribution functions; and

  2. 2.

    Representing and operating on operands when one is a distribution function and the other is an interval.

We discuss how to do these operations and get automatically verified results, using a method based on histograms. Histograms are composed of bars, each with an associated interval and probability mass. A histogram discretizes a probability distribution function in that each bar represents a portion of it, with the area of the bar equal to the area of the distribution function over that part of its domain corresponding to the interval-valued portion of the x-axis under the bar.

This is a preview of subscription content, 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   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Ahmed, R. E. Clark, and D. R. Metcalf, “A Method for Propagating Uncertainty in Probabilistic Risk Assessment,” Nuclear Technology, 1982, Vol. 59, pp. 238–245.

    Google Scholar 

  2. D. Berleant, “Automatically Verified Reasoning with Both Intervals and Probability Density Functions,” Interval Computations, 1993, No. 2, pp. 48–70.

    MathSciNet  Google Scholar 

  3. H. Cheng and D. Berleant, “A Software Tool for Automatically Verified Operations on Intervals and Probability Distributions,” submitted to the International Journal Reliable Computing.

    Google Scholar 

  4. A. G. Colombo and R. J. Jaarsma, “A Powerful Numerical Method to Combine Random Variables”, IEEE Transactions on Reliability, 1980, Vol. R-29, No. 2, pp. 126–129.

    Article  Google Scholar 

  5. F. Corsi, “Mathematical Models for Marginal Reliability Analysis,” Microelectronics and Reliability, 1983, Vol. 23, No. 6, pp. 1087–1102.

    Article  Google Scholar 

  6. V. A. Gerasimov, B. S. Dobronets, and M. Yu. Shustrov, “Numerical Calculations for Histogram Operations and their Applications,” Avtomatika i Telemekhanika, 1991, Vol. 52, No. 2.

    Google Scholar 

  7. E. Hyvönen, Constraint Reasoning on Probability Distributions, manuscript (1995), author is at VTT Information Technology, P.O. Box 1201, 02044 VTT, Finland.

    Google Scholar 

  8. G. E. Ingram, E. L. Welker, and C. R. Herrmann, “Designing for Reliability Based on Probabilistic Modeling Using Remote Access Computer Systems”, Proceedings 7th Reliability and Maintainability Conference, American Society of Mechanical Engineers, 1968, pp. 492–500.

    Google Scholar 

  9. P. S. Jackson, R. W. Hockenbury, and M. L. Yeater, “Uncertainty Analysis of System Reliability and Availability Assessment”, Nuclear Engineering and Design, 1981, Vol. 68, 1981, pp. 5–29.

    Google Scholar 

  10. S. Kaplan, “On the Method of Discrete Probability Distributions in Risk and Reliability Calculations — Application to Seismic Risk Assessment,” Risk Analysis, 1981, Vol. 1, No. 3, pp. 189–196.

    Article  Google Scholar 

  11. A. Kolmogoroff (a. k. a. Kolmogorov), “Confidence Limits for an Unknown Distribution Function,” Annals of Mathematical Statistics, 1941, Vol. 12, No. 4, pp. 461–463.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Levy, “Stochastic Dominance and Expected Utility: Survey and Analysis,” Management Science, 1992, Vol. 38, No. 4, pp. 555–593.

    Article  MATH  Google Scholar 

  13. A. S. Moore, “Interval Risk Analysis of Real Estate Investment: A Non-Monte Carlo Approach,” Freiburger Intervall-Berichte, 1985, Vol. 85 /3, pp. 23–49.

    Google Scholar 

  14. R. E. Moore, “Risk Analysis Without Monte Carlo Methods,” Freiburger Intervall-Berichte, 1984, Vol. 84 /1, pp. 1–48.

    Google Scholar 

  15. G. V. Post and J. D. Diltz, “A Stochastic Dominance Approach to Risk Analysis of Computer Systems,” Management Information Systems Quarterly, 1986, Vol. 10, pp. 363–375.

    Article  Google Scholar 

  16. A. M. Rushdi and K. F. Kafrawy, “Uncertainty Propagation in Fault-Tree Analyses Using an Exact Method of Moments,” Microelectronics and Reliability, 1988, Vol. 28, No. 6, pp. 945–965.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Systems Engineering, University of Arkansas, Fayetteville, AR, 72701, USA

    Daniel Berleant

Authors
  1. Daniel Berleant
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Southwestern Louisiana, USA

    R. Baker Kearfott

  2. University of Texas at El Paso, USA

    Vladik Kreinovich

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Kluwer Academic Publishers

About this chapter

Cite this chapter

Berleant, D. (1996). Automatically Verified Arithmetic on Probability Distributions and Intervals. In: Kearfott, R.B., Kreinovich, V. (eds) Applications of Interval Computations. Applied Optimization, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3440-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-3440-8_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3442-2

  • Online ISBN: 978-1-4613-3440-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation