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A characterization of the negative exponential distribution with application to reliability theory

Published online by Cambridge University Press:  14 July 2016

M. J. Phillips
M. J. Phillips*
Affiliation:
University of Leicester
*
∗ Postal address: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, U.K.
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Abstract

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

Keywords

CHARACTERIZATION THEOREMNEGATIVE EXPONENTIAL DISTRIBUTIONRELIABILITYREVEALED AND UNREVEALED FAULTSALTERNATING RENEWAL PROCESSTWO-STATE SEMI-MARKOV PROCESSSYSTEM AVAILABILITY

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981, pp. 652 - 659
Copyright
Copyright © Applied Probability Trust 

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References

Basu, A. P. (1965) On characterizing the exponential distribution by order statistics. Ann. Inst. Statist. Math. 17, 93–96.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Crawford, G. B. (1966) Characterization of geometric and exponential distributions. Ann. Math. Statist. 37, 1790–1795.Google Scholar
Desu, M. M. (1971) A characterization of the exponential distribution by order statistics. Ann. Math. Statist. 42, 837–838.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications, Vol. I. Wiley, New York.Google Scholar
Ferguson, T. S. (1964) A characterization of the exponential distribution. Ann. Math. Statist 35, 1199–1207.Google Scholar
Galambos, J. (1975) Characterizations in terms of properties of the smaller of two observations. Commun. Statist. 4, 239–244.CrossRefGoogle Scholar
Govindarajulu, Z. (1966) Characterization of the exponential and power distributions. Skand. Aktuarietidskr. 49, 132–136.Google Scholar
Kagan, A. M., Linnik, Yu.V. and Rao, C. R. (1973) Characterization Problems in Mathematical Statistics. Wiley, New York.Google Scholar
Kotz, S. (1974) Characterizations of statistical distributions: a supplement to recent surveys. Internat. Statist. Rev. 42, 39–65.Google Scholar
Krishnaji, N. (1971) Note on a characterizing property of the exponential distribution. Ann. Math. Statist. 42, 361–362.Google Scholar
Murphy, T. (1968) An Account of Renewal Theory with Illustrations of its Application to Reliability and Safety Assessment. AHSB(S)R. 145 U.K.A.E.A., Warrington, Lancashire.Google Scholar
Muth, E. J. (1977) Transform Methods with Applications to Engineering and Operations Research. Prentice-Hall, Englewood Cliffs, N. J. Google Scholar
Phillips, M. J. (1979) The reliability of a system subject to revealed and unrevealed faults. Microelectron Reliab. 18, 495–503.Google Scholar
Puri, P. S. (1973) On a property of exponential and geometric distributions and its relevance to multivariate failure rate. Sankhya A 35, 61–68.Google Scholar
Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
Radner, R. and Jorgensen, D. W. (1962) Optimal replacement and inspection of stochastically failing equipment. In Studies in Applied Probability and Management Science, ed. Arrow, K. J. et al. Stanford University Press, Stanford.Google Scholar
Radner, R. and Jorgensen, D. W. (1963) Opportunistic replacement of a single part in the presence of several monitored parts. Management Sci. 10, 70–82.CrossRefGoogle Scholar
Ramachandran, B. (1977) On the strong Markov property of the exponential laws. Proceedings of the Colloquium on the Methods of Complex Analysis in the Theory of Probability and Statistics. Debrecen, Hungary.Google Scholar
Reinhardt, H. E. (1968) Characterizing the exponential distribution. Biometrics 24, 437–439.Google Scholar
Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden Day, San Francisco.Google Scholar
Shanbhag, D. N. (1970) The characterizations for exponential and geometric distributions. J. Amer. Statist. Assoc. 65, 1256–1259.Google Scholar
Shimizu, R. (1978) Solution to a functional equation and its application to some characterization problems. Sankhya A 40, 319–332.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
Srivastava, M. S. (1967) A characterization of the exponential distribution. Amer. Math. Monthly 74, 414–416.Google Scholar
Tanis, E. A. (1964) Linear forms in the order statistics from an exponential distribution. Ann. Math. Statist. 35, 270–276.Google Scholar