The exponential distribution is often characterized as the only distribution with lack of memory. This note points out a stronger result: the exponential is the only distribution that is occasionally forgetful.
"A Note on the "Lack of Memory" Property of the Exponential Distribution." Ann. Probab. 3 (2) 353 - 354, April, 1975. https://doi.org/10.1214/aop/1176996406