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 is the waiting time from today until the next earthquake, and . Suppose the world doesn’t have an earthquake for 3 months.
This does not make us due for an earthquake.
If we’ve waited of a year, then the additional waiting time left is given by . However, we know that , as we’ve already passed 3 months. Then, the distribution is given by
Here, 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 .
It doesn’t matter how long we’ve “waited.” An additional wait length 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. from that distribution satisfies
for all . Furthermore, these two distributions are Independent of one another.