Editing Memorylessness (section)

== Characterization of exponential distribution ==
If a continuous probability distribution is memoryless, then it must be the exponential distribution.

From the memorylessness property,<math display="block">\Pr(X>t+s \mid X>s)=\Pr(X>t).</math>The definition of [[conditional probability]] reveals that<math display="block">\frac{\Pr(X > t + s)}{\Pr(X > s)} = \Pr(X > t).</math>Rearranging the equality with the [[survival function]], <math>S(t) = \Pr(X > t)</math>, gives<math display="block">S(t + s) = S(t) S(s).</math>This implies that for any [[natural number]] <math>k</math><math display="block">S(kt) = S(t)^k.</math>Similarly, by dividing the input of the survival function and taking the <math>k</math>-th root,<math display="block">S\left(\frac{t}{k}\right) = S(t)^{\frac{1}{k}}.</math>In general, the equality is true for any [[rational number]] in place of <math>k</math>. Since the survival function is [[Continuous function|continuous]] and rational numbers are [[Dense set|dense]] in the [[Real number|real numbers]] (in other words, there is always a rational number arbitrarily close to any real number), the equality also holds for the reals. As a result,<math display="block">S(t) = S(1)^t = e^{t \ln S(1)} = e^{-\lambda t}</math>where <math>\lambda = -\ln S(1) \geq 0</math>. This is the survival function of the exponential distribution.<ref name=":2" />