7.2.1 Lack of memory

A key property of the exponential distribution is its lack of memory property.

A random variable satisfies the lack of memory property if $$\mathrm{P}(X>s+t\mid X>t)=\mathrm{P}(X>s)$$ for $s>0,t>0$ i.e. the conditional probability that a variable exceeds $s+t$, given that it exceeds $t$, is independent of $t$ so it has no memory for how large it is already.

If we interpret $X$ as a waiting time to an event, this means that the probability that you have to wait a further time $s$ is independent of how long you have waited already.

To show this result holds for $X\sim \mathrm{Exp}(\beta )$ recall that $\mathrm{P}(X>x)=\exp (-\beta x)$ for all $x>0$. Hence, for $s>0$, $t>0$

Note that we already observed this phenomenon in Example 7.7

Actually, one can show that the Exponential distribution is the only continuous distribution with the lack of memory property. We will not show it here.