FM 2009: FM 2009: Formal Methods pp 435-450 | Cite as
Formal Reasoning about Expectation Properties for Continuous Random Variables
Abstract
Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higher-order-logic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation of a continuous random variable in terms of simple arithmetic operations. In order to illustrate the practical effectiveness and utilization of our approach, we also present the formal verification of expectation properties of the commonly used continuous random variables: Uniform, Triangular and Exponential.
Keywords
Theorem Prover Real Sequence Discrete Random Variable Continuous Random Variable Formal ReasoningPreview
Unable to display preview. Download preview PDF.
References
- 1.Akbarpour, B., Tahar, S.: An Approach for the Formal Verification of DSP Designs using Theorem Proving. IEEE Transactions on CAD of Integrated Circuits and Systems 25(8), 1141–1457 (2006)Google Scholar
- 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.Bialas, J.: The σ-Additive Measure Theory. J. of Formalized Mathematics 2 (1990)Google Scholar
- 4.Coble, A.: On Probability, Measure, and Integration in HOL4. Technical Report, Computing Laboratory, University of Cambridge, UK (2009), http://www.srcf.ucam.org/~arc54/techreport.pdf
- 5.Daumas, M., Martin-Dorel, E., Lester, D., Truffert, A.: Stochastic Formal Correctness of Numerical Allgorithms. In: First NASA Formal Methods Symposium, pp. 136–145 (2009)Google Scholar
- 6.Devroye, L.: Non-Uniform Random Variate Generation. Springer, Heidelberg (1986)MATHGoogle Scholar
- 7.Galambos, J.: Advanced Probability Theory. Marcel Dekker Inc., New York (1995)MATHGoogle Scholar
- 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.Harrison, J.: Floating Point Verification in HOL Light: The Exponential Function. Technical Report 428, Computing Laboratory, University of Cambridge, UK (1997)Google Scholar
- 10.Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)MATHGoogle Scholar
- 11.Hasan, O.: Formal Probabilistic Analysis using Theorem Proving. PhD Thesis, Concordia University, Montreal, QC, Canada (2008)Google Scholar
- 12.Hasan, O., Abbasi, N., Tahar, S.: Formal Probabilistic Analysis of Stuck-at Faults in Reconfigurable Memory Arrays. In: Leuschel, M., Wehrheim, H. (eds.) IFM 2009. LNCS, vol. 5423, pp. 277–291. Springer, Heidelberg (2009)CrossRefGoogle Scholar
- 13.Hasan, O., Tahar, S.: Performance Analysis of ARQ Protocols using a Theorem Prover. In: Proc. International Symposium on Performance Analysis of Systems and Software, pp. 85–94. IEEE Computer Society, Los Alamitos (2008)CrossRefGoogle Scholar
- 14.Hasan, O., Tahar, S.: Performance Analysis of Wireless Systems using Theorem Proving. In: Proc. First International Workshop on Formal Methods for Wireless Systems, Toronto, ON, Canada, pp. 3–18 (2008)Google Scholar
- 15.Hurd, J.: Formal Verification of Probabilistic Algorithms. PhD Thesis, University of Cambridge, Cambridge, UK (2002)Google Scholar
- 16.Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
- 17.Nedzusiak, A.: σ-fields and Probability. J. of Formalized Mathematics 1 (1989)Google Scholar
- 18.Richter, S.: Formalizing Integration Theory, with an Application to Probabilistic Algorithms. Diploma Thesis, Technische Universität München, Department of Informatics, Germany (2003)Google Scholar
- 19.Widrow, B.: Statistical Analysis of Amplitude-quantized Sampled Data Systems. AIEE Trans. (Applications and Industry) 81, 555–568 (1961)Google Scholar