Editing Memorylessness

{{Short description|Waiting time property of certain probability distributions}}
{{About||use of the term in [[materials science]]|hysteresis|use of the term in [[stochastic process]]es and [[Markov chain]]s|Markov property}}

In [[probability]] and [[statistics]], '''memorylessness''' is a property of [[probability distribution]]s. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the [[geometric distribution|geometric]] and [[exponential distribution|exponential]] distributions are memoryless.

== 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>

== Characterization of exponential distribution ==
If a continuous probability distribution is memoryless, then it must be the exponential distribution.

From the memorylessness property,<math display="block">\Pr(X>t+s \mid X>s)=\Pr(X>t).</math>The definition of [[conditional probability]] reveals that<math display="block">\frac{\Pr(X > t + s)}{\Pr(X > s)} = \Pr(X > t).</math>Rearranging the equality with the [[survival function]], <math>S(t) = \Pr(X > t)</math>, gives<math display="block">S(t + s) = S(t) S(s).</math>This implies that for any [[natural number]] <math>k</math><math display="block">S(kt) = S(t)^k.</math>Similarly, by dividing the input of the survival function and taking the <math>k</math>-th root,<math display="block">S\left(\frac{t}{k}\right) = S(t)^{\frac{1}{k}}.</math>In general, the equality is true for any [[rational number]] in place of <math>k</math>. Since the survival function is [[Continuous function|continuous]] and rational numbers are [[Dense set|dense]] in the [[Real number|real numbers]] (in other words, there is always a rational number arbitrarily close to any real number), the equality also holds for the reals. As a result,<math display="block">S(t) = S(1)^t = e^{t \ln S(1)} = e^{-\lambda t}</math>where <math>\lambda = -\ln S(1) \geq 0</math>. This is the survival function of the exponential distribution.<ref name=":2" />

== Characterization of geometric distribution ==
If a discrete probability distribution is memoryless, then it must be the geometric distribution.

From the memorylessness property,<math display="block">\Pr(X>t+s \mid X\geq s)=\Pr(X>t)</math>The definition of [[conditional probability]] reveals that<math display="block">\frac{\Pr(X > t + s)}{\Pr(X \geq s)} = \Pr(X > t)</math>From this it can be proven by induction that <math display="block">\Pr(X > kt) = \Pr(X > 1)^k</math>Then it follows that<math display="block">f_X(x)=Pr(X\leq x)=1-Pr(X>x)=1-Pr(X>1)^x</math> and if we let <math display="block">Pr(X>1)=1-p</math>for some <math>0\leq p \leq 1</math>. we can easily see that X is geometrically distributed with some parameter p. in other words
<math display="block">X\sim Geo(p)</math>

== References ==
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[[Category:Theory of probability distributions]]
[[Category:Characterization of probability distributions]]
[[Category:Articles containing proofs]]