Editing Memorylessness (section)

== Definition ==
A [[random variable]] <math>X</math> is memoryless if <math display="block">\Pr(X>t+s \mid X>s)=\Pr(X>t)</math>where <math>\Pr</math> is its [[probability mass function]] or [[probability density function]] when <math>X</math> is [[Discrete random variable|discrete]] or [[Continuous random variable|continuous]] respectively and <math>t</math> and <math>s</math> are [[nonnegative]] numbers.<ref name=":1">{{Cite book |last=Dekking |first=Frederik Michel |url=http://link.springer.com/10.1007/1-84628-168-7 |title=A Modern Introduction to Probability and Statistics |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |publisher=Springer London |year=2005 |isbn=978-1-85233-896-1 |series=Springer Texts in Statistics |location=London |page=50 |doi=10.1007/1-84628-168-7}}</ref><ref>{{Cite book |last=Pitman |first=Jim |url=http://link.springer.com/10.1007/978-1-4612-4374-8 |title=Probability |publisher=Springer New York |year=1993 |isbn=978-0-387-94594-1 |location=New York, NY |page=279 |language=en |doi=10.1007/978-1-4612-4374-8}}</ref> In discrete cases, the definition describes the first success in an infinite sequence of [[Independent and identically distributed random variables|independent and identically distributed]] [[Bernoulli trial|Bernoulli trials]], like the number of coin flips until landing heads.<ref>{{Cite book |last=Nagel |first=Werner |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496 |title=Probability and Conditional Expectation: Fundamentals for the Empirical Sciences |last2=Steyer |first2=Rolf |date=2017-04-04 |publisher=Wiley |isbn=978-1-119-24352-6 |edition=1st |series=Wiley Series in Probability and Statistics |pages=260–261 |language=en |doi=10.1002/9781119243496}}</ref> In continuous situations, memorylessness models random phenomena, like the time between two earthquakes.<ref>{{Cite book |last=Bas |first=Esra |url=http://link.springer.com/10.1007/978-3-030-32323-3 |title=Basics of Probability and Stochastic Processes |publisher=Springer International Publishing |year=2019 |isbn=978-3-030-32322-6 |location=Cham |page=74 |language=en |doi=10.1007/978-3-030-32323-3}}</ref> The memorylessness property asserts that the number of previously failed trials or the elapsed time is [[Independence (probability theory)|independent]], or has no effect, on the future trials or lead time.

The equality [[Characterization (mathematics)|characterizes]] the [[geometric distribution|geometric]] and [[Exponential distribution|exponential distributions]] in discrete and continuous contexts respectively.<ref name=":1" /><ref name=":2">{{Cite book |last=Riposo |first=Julien |url=https://link.springer.com/10.1007/978-3-031-31323-3 |title=Some Fundamentals of Mathematics of Blockchain |publisher=Springer Nature Switzerland |year=2023 |isbn=978-3-031-31322-6 |location=Cham |pages=8–9 |language=en |doi=10.1007/978-3-031-31323-3}}</ref> In other words, the geometric random variable is the only discrete memoryless distribution and the exponential random variable is the only continuous memoryless distribution.

In discrete contexts, the definition is altered to <math display="inline">\Pr(X>t+s \mid X \geq s)=\Pr(X>t)</math> when the geometric distribution starts at <math>0</math> instead of <math>1</math> so the equality is still satisfied.<ref>{{Cite book |last=Johnson |first=Norman L. |url=https://onlinelibrary.wiley.com/doi/book/10.1002/0471715816 |title=Univariate Discrete Distributions |last2=Kemp |first2=Adrienne W.|author2-link=Adrienne W. Kemp |last3=Kotz |first3=Samuel |date=2005-08-19 |publisher=Wiley |isbn=978-0-471-27246-5 |edition=1 |series=Wiley Series in Probability and Statistics |pages=210 |language=en |doi=10.1002/0471715816}}</ref><ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |last2=Ross |first2=Andrew M. |title=Memoryless |url=https://mathworld.wolfram.com/Memoryless.html |url-status=live |archive-url=https://web.archive.org/web/20241202153603/https://mathworld.wolfram.com/Memoryless.html |archive-date=2024-12-02 |access-date=2024-07-25 |website=mathworld.wolfram.com |language=en}}</ref>