among all continuous ones. The property is derived through the following proof: To see this, first define the survival function, , as Note that is then monotonically decreasing. From the relation and the definition of conditional probability, it follows that This gives the functional equation (which is a result of the memorylessness property): From this, we must have for example: In general: The only continuous function that will satisfy this equation for any positive, rational is: where Therefore, since is a probability and must have then any memorylessness function must be an exponential. Put a different way, is a monotone decreasing function (meaning that for times then ) The functional equation alone will imply that restricted to rational multiples of any particular number is an exponential function. Combined with the fact that is monotone, this implies that over its whole domain is an exponential function. Notes References Feller, W. (1971) Introduction to Probability Theory and Its Applications, Vol II (2nd edition),Wiley. Section I.3 Theory of probability distributions Characterization of probability distributions Markov processes