Editing Exponential distribution (section)

===Memorylessness property of exponential random variable===
An exponentially distributed random variable ''T'' obeys the relation
<math display="block">\Pr \left (T > s + t \mid T > s \right ) = \Pr(T > t), \qquad \forall s, t \ge 0.</math>

This can be seen by considering the [[complementary cumulative distribution function]]:
<math display="block"> \begin{align}
  \Pr\left(T > s + t \mid T > s\right)
    &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\[4pt]
    &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\[4pt]
    &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\[4pt]
    &= e^{-\lambda t} \\[4pt]
    &= \Pr(T > t).
\end{align} </math>

When ''T'' is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if ''T'' is conditioned on a failure to observe the event over some initial period of time ''s'', the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the [[conditional probability]] that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds after the initial time.

The exponential distribution and the [[geometric distribution]] are [[memorylessness|the only memoryless probability distributions]].

The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant [[failure rate]].