In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property $$ P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right], $$ where $a,t\ge 0$. We noticed some interesting properties that follow, and wonder if the above property is known in the community. Notice that in the case of equality, this is equivalent to the memoryless property.
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1$\begingroup$ To me, this seems related to increasing/decreasing failure rates (and their variants) that are classically used in reliability analysis. I do not know the name of your property but I think that it probably has one. $\endgroup$– N. GastCommented Apr 11, 2019 at 14:18
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$\begingroup$ Have a look in Barlow/Proschan (1996), Mathematical Theory of Reliability, Chapter 2, $\endgroup$– Dieter KadelkaCommented Apr 11, 2019 at 15:27
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1$\begingroup$ A similar term to failure rate is "hazard rate", both appear in this wikipedia article. en.wikipedia.org/wiki/Failure_rate#Decreasing_failure_rate $\endgroup$– usulCommented Apr 12, 2019 at 0:19
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By googling
"decreasing conditional survival" function
we find this term used e.g. in this paper and a few others.
answered Apr 11, 2019 at 15:33