The property possessed by the Poisson process, and by its discrete analogue (a sequence of Bernoulli trials), that probabilities of future occurrences are uninfluenced by past events. This is just a restatement of independence. The corresponding distributions of the time to the first observation (either exponential or geometric distributions) and their extensions to the time to the nth observation (either gamma or negative binomial distributions) all have the forgetfulness property.
Forgetfulness property. For the exponential distribution, if we know that the observed value is greater than a, then the resulting conditional distribution is a facsimile of the original distribution.
Subjects: Probability and Statistics.