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On a “lack of memory” property

  • J. S. Huang
Article

Abstract

For two independent nonnegative random variablesX andY we say thatX is ageless relative toY if the conditional probability P[X> Y+x|X>Y] is defined and is equal to P[X>x] for allx>0. Suppose thatX is ageless relative to a nonlatticeY with P[Y=0]<P [Y<X]. We show that the only suchX is the exponential variable. As a corollary it follows that exponential variable is the only one which possesses the ageless property relative to a continuous variable.

AMS 1970 subject classifications

60E05 60H20 62E10 

Key words

Exponential distribution characterization lack of memory nonlattice distribution Laplace transform analytic continuation Wiener-Hopf technique 

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References

  1. [1]
    Aczel, J. (1966).Lectures on Functional Equations and Their Applications, Academic Press, New York.zbMATHGoogle Scholar
  2. [2]
    Barlow, R. and Proschan, F. (1975).Statistical theory of reliability and life testing probability models, Holt, Rinehart and Winston, New York.zbMATHGoogle Scholar
  3. [3]
    Cinlar, E. and Jagers, P. (1973). Two mean values which characterize the Poisson process,J. Appl. Prob.,10, 678–681.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Feller, W. (1967).An introduction to probability theory and its applications, 1, 3rd ed., Wiley, New York.zbMATHGoogle Scholar
  5. [5]
    Krishnaji, N. (1970). Characterization of the Pareto distribution through a model of underreported incomes,Econometrica,38, 251–255.CrossRefGoogle Scholar
  6. [6]
    Krishnaji, N. (1971). Note on a characterizing property of the exponential distribution,Ann. Math. Statist.,42, 361–362.CrossRefGoogle Scholar
  7. [7]
    Marsaglia, G. and Tubilla, A. (1975). A note on the “lack of memory” property of the exponential distribution,Ann. Prob.,3, 353–354.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Obreteno, A. (1970). A property of the exponential distribution (Bulgarian),Fiz.-Mat. Spis. B″ lgar. Akad. Nauk.,13 (46), 51–53.Google Scholar
  9. [9]
    Ramachandran, B. (1977). On the strong Markov property of the exponential laws, paper contributed to theColloquium on the Methods of Complex Analysis in the Theory of Probability and Statistics, held at Debrecen, Hungary, Aug.-Sept.Google Scholar
  10. [10]
    Rossberg, H.-J. (1972). Characterization of the exponential and the Pareto distributions by means of some properties of the distributions which the differences and quotients of order statistics are subject to,Math. Operat. u. Statist.,3, 207–216.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Shimizu, R. (1978). Solution to a functional equation and its application to some characterization problems,Sankhyā, A,40, 319–332.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Widder, D. (1946).The Laplace transform, Princeton Univ. Press, Princeton.zbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1981

Authors and Affiliations

  • J. S. Huang
    • 1
  1. 1.University of GuelphGuelphCanada

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