---
tag: 0ZAZZ34WQS
kind: concept
name: Memorylessness
aliases: [memoryless-property, lack-of-memory]
msc: ["60E05"]
uses: [0FJB06AD2P, 028Z8WC5Y9, 0XCVV5Q500]
sources:
  - url: https://en.wikipedia.org/wiki/Memorylessness
    locator: Definition
same_as:
  wikidata: Q176750
  wikipedia: Memorylessness
---

## Summary

A random waiting time $X \ge 0$ is memoryless if, for all $s, t \ge 0$ with $P(X > s) > 0$,
$$P(X > s + t \mid X > s) = P(X > t).$$
So the time already waited does not change the law of the remaining waiting time.
Among the distributions on $[0, \infty)$ with $P(X > t) > 0$ for each $t$, only the [[exponential distributions|028Z8WC5Y9]] are memoryless. Among the distributions on $\{0, 1, 2, \dots\}$, with $s$ and $t$ integers, only the [[geometric distributions|0XCVV5Q500]] are memoryless.

## Remarks

For $X \sim \operatorname{Exp}(\lambda)$, $P(X > s + t) = e^{-\lambda(s + t)} = P(X > s) \, P(X > t)$. This is the memoryless property.
The proof of the characterization uses that $G(t) = P(X > t)$ satisfies $G(s + t) = G(s) \, G(t)$ and is right-continuous. The only such functions with values in $(0, 1]$ are $G(t) = e^{-\lambda t}$ with $\lambda \ge 0$.

## Explanations

### L1 human

A memoryless waiting time does not "age".
If a bus comes at exponential times and you have already waited ten minutes, your expected further wait is the same as when you arrived.