The memoryless distribution is an exponential distribution
The only memoryless continuous probability distribution is the exponential distribution, so memorylessness completely characterizes the exponential distribution among all continuous ones. The property is derived through the following proof:
To see this, first define the survival function, S, as
Note that S(t) 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 a is:
where
Therefore, since S(a) is a probability and must have , then any memorylessness function must be an exponential.
Put a different way, S is a monotone decreasing function (meaning that for times then )
The functional equation alone will imply that S restricted to rational multiples of any particular number is an exponential function. Combined with the fact that S is monotone, this implies that S over its whole domain is an exponential function.