# On a “lack of memory” property

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## Abstract

For two independent nonnegative random variables*X* and*Y* we say that*X* is ageless relative to*Y* if the conditional probability P[*X> Y+x|X>Y*] is defined and is equal to P[*X>x*] for all*x*>0. Suppose that*X* is ageless relative to a nonlattice*Y* with P[*Y*=0]<P [*Y<X*]. We show that the only such*X* 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## Preview

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© The Institute of Statistical Mathematics, Tokyo 1981