Let be a positive random variable which is not identically zero or . Then has the exponential distribution if and only if has the memoryless property.

Necessity is the computation in the proposition above. For sufficiency, suppose has the memoryless property and set

Then , so for every ; taking , . Writing gives for all , and from we get . Hence for all rational .

To extend to , let and choose rationals with . Since is decreasing,

and letting yields : the survival function of .