An integrated lack of memory characterization of the exponential distribution

Abstract

SummaryIn this note a characterization of the exponential distribution is discussed based on yet another extension of the lack of memory property. The result was motivated by a functional equation appearing in Ahsannulah [1], [3].

A method associated with characterizations of the exponential distribution

“…Hence, if the function g(z) satisfies equation (3), then g(z) is differentiable and its derivative g'(z) also satisfies equation (3). Thus, g(z) is an infinitely differentiable function (for z>0) and also all of its derivatives satisfy (3).…”

“…In particular, we may make no assumption on the positiveness of the function f(x). Besides, our regularity assumptions are different from those of [3]. In the papers by Grosswald, Kotz and Johnson [4] and Grosswald and Kotz [3] a related relevation type equation…”

“…of the exponential law. Grosswald and Kotz [3] studied equation (1) under somewhat different regularity assumptions. However the assumptions of [3] are, nevertheless, rather restrictive and, besides, there is considered only the case when f(x) is the density function of F(x).…”