Think of this as the “continuous” version of “if I flip a coin 100 times and get 50 heads, then the next one must be tails, because we’re due for one.” *It is incorrect.*

Example

Support $X$ is the waiting time from today until the next earthquake, and $X∼Expo(17)$. Suppose the world doesn’t have an earthquake for 3 months.

This does not

make us due for an earthquake.If we’ve waited $41 $ of a year, then the additional waiting time left is given by $X−41 $. However, we know that $X≥41 $, as we’ve already passed 3 months. Then, the distribution is given by

$P(X−41 ∣X≥41 )=P(X≥t)$Here, $X≥41 $ is the event that we’re still waiting for the next earthquake, 3 months down the line. The distribution is the same as the original distribution of $X$.

It doesn’t matter how long we’ve “waited.” An additional wait length $t$ is just as likely now as it was when we first started.

- The Exponential distribution is the only continuous, memoryless distribution.
- Realistic for some processes (radioactive decay), less so for others (human life).
- Connected to Poisson processes and the “finite” version of the Geometric distribution, which counts the “finite” waiting time until the first success.

## Formal definition

Formally, we say a distribution is **memoryless** if a r.v. $X$ from that distribution satisfies

for all $s,t>0$. Furthermore, these two distributions are Independent of one another.