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 are memoryless. Among the distributions on $\{0, 1, 2, \dots\}$ , with $s$ and $t$ integers, only the geometric distributions 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 $\lambda \ge 0$ .

Tag	`0ZAZZ34WQS`
URI	`m2://0ZAZZ34WQS`
Aliases	`memoryless-property`, `lack-of-memory`
Kind	concept
Writing	W0 stub
MSC	60E05
Uses	Conditional probability, Exponential distribution, Geometric distribution
Same as	Wikidata Q176750 Wikipedia Memorylessness

Tag

0ZAZZ34WQS

URI

m2://0ZAZZ34WQS

Aliases

memoryless-property, lack-of-memory

Kind

concept

Writing

W0 stub

MSC

60E05

Uses

Conditional probability, Exponential distribution, Geometric distribution

Same as

Wikidata Q176750

Wikipedia Memorylessness

Memorylessness

Summary

Remarks

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