Formalization of Continuous Probability Distributions

  • Osman Hasan
  • Sofiène Tahar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4603)


Continuous probability distributions are widely used to mathematically describe random phenomena in engineering and physical sciences. In this paper, we present a methodology that can be used to formalize any continuous random variable for which the inverse of the cumulative distribution function can be expressed in a closed mathematical form. Our methodology is primarily based on the Standard Uniform random variable, the classical cumulative distribution function properties and the Inverse Transform method. The paper includes the higher-order-logic formalization details of these three components in the HOL theorem prover. To illustrate the practical effectiveness of the proposed methodology, we present the formalization of Exponential, Uniform, Rayleigh and Triangular random variables.


Guaran Veri 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akbarpour, B., Tahar, S.: Formalization of Fixed-Point Arithmetic in HOL. Formal Methods in Systems Design 27(1-2), 173–200 (2005)MATHCrossRefGoogle Scholar
  2. 2.
    Audebaud, P., Paulin-Mohring, C.: Proofs of Randomized Algorithms in Coq. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 49–68. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P: Model Checking Algorithms for Continuous time Markov Chains. IEEE Trans. on Software Engineering 29(4), 524–541 (2003)CrossRefGoogle Scholar
  4. 4.
    Clarke, E.M, Grumberg, O., Peled, D.A: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  5. 5.
    Devroye, L.: Non-Uniform Random Variate Generation. Springer, Heidelberg (1986)MATHGoogle Scholar
  6. 6.
    Gonsalves, T.A, Tobagi, F.A: On the Performance Effects of Station Locations and Access Protocol Parameters in Ethernet Networks. IEEE Trans. on Communications 36(4), 441–449 (1988)CrossRefGoogle Scholar
  7. 7.
    Gordon, M.J.C: Mechanizing Programming Logics in Higher-0rder Logic. In: Current Trends in Hardware Verification and Automated Theorem Proving, pp. 387–439. Springer, Heidelberg (1989)Google Scholar
  8. 8.
    Gordon, M.J.C, Melham, T.F: Introduction to HOL: A Theorem Proving Environment for Higher-Order Logic. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  9. 9.
    Gupta, V.T, Jagadeesan, R., Panangaden, P.: Stochastic Processes as Concurrent Constraint Programs. In: Principles of Programming Languages, pp. 189–202. ACM Press, New York (1999)Google Scholar
  10. 10.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)MATHGoogle Scholar
  11. 11.
    Hasan, O., Tahar, S.: Formalization of the Standard Uniform Random Variable. Theoretical Computer Science (to appear)Google Scholar
  12. 12.
    Hasan, O., Tahar, S.: Verification of Probabilistic Properties in HOL using the Cumulative Distribution Function. In: Integrated Formal Methods. LNCS, vol. 4591, pp. 333–352. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Hasan, O., Tahar, S.: Formalization of Continuous Probability Distributions. Technical Report, Concordia University, Montreal, Canada (February 2007),
  14. 14.
    Hurd, J.: Formal Verification of Probabilistic Algorithms. PhD Thesis, University of Cambridge, Cambridge, UK (2002)Google Scholar
  15. 15.
    Kaneko, T., Liu, B.: On Local Roundoff Errors in Floating-Point Arithmetic. ACM 20(3), 391–398 (1973)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Khazanie, R.: Basic Probability Theory and Applications. Goodyear (1976)Google Scholar
  17. 17.
    Köpsel, A., Ebert, J., Wolisz, A.: A Performance Comparison of Point and Distributed Coordination Function of an IEEE 802.11 WLAN in the Presence of Real-Time Requirements. In: Proceedings of Seventh International Workshop on Mobile Multimedia Communications, Tokyo, Japan (2000)Google Scholar
  18. 18.
    Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  19. 19.
    Park, S., Pfenning, F., Thrun, S.: A Probabilistic Language based upon Sampling Functions. In: Principles of Programming Languages, pp. 171–182. ACM Press, New York (2005)Google Scholar
  20. 20.
    Pfeffer, A.: IBAL: A Probabilistic Rational Programming Language. In: International Joint Conferences on Artificial Intelligence, pp. 733–740. Morgan Kaufmann Publishers, Washington (2001)Google Scholar
  21. 21.
    Ross, S.M: Simulation. Academic Press, San Diego (2002)Google Scholar
  22. 22.
    Rutten, J., Kwaiatkowska, M., Normal, G., Parker, D.: Mathematical Techniques for Analyzing Concurrent and Probabilisitc Systems. CRM Monograph Series. American Mathematical Society, vol. 23 (2004)Google Scholar
  23. 23.
    Tridevi, K.S: Probability and Statistics with Reliability, Queuing and Computer Science Applications. Wiley, Chichester (2002)Google Scholar
  24. 24.
    Widrow, B.: Statistical Analysis of Amplitude-quantized Sampled Data Systems. AIEE Trans. (Applications and Industry) 81, 555–568 (1961)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Osman Hasan
    • 1
  • Sofiène Tahar
    • 1
  1. 1.Dept. of Electrical & Computer Engineering, Concordia University, 1455 de Maisonneuve W., Montreal, Quebec, H3G 1M8Canada

Personalised recommendations