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<title>Memorylessness</title>

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Memorylessness

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</style><div role="note" class="hatnote navigation-not-searchable">For use of the term in <a href="Materials_science" title="Materials science">materials science</a>, see <a href="Hysteresis" title="Hysteresis">hysteresis</a>. For use of the term in <a href="Stochastic_process" title="Stochastic process">stochastic processes</a> and <a href="Markov_chain" title="Markov chain">Markov chains</a>, see <a href="Markov_property" title="Markov property">Markov property</a>.</div>

In <a href="Probability" title="Probability">probability</a> and <a href="Statistics" title="Statistics">statistics</a>, memorylessness is a property of <a href="Probability_distribution" title="Probability distribution">probability distributions</a>. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the <a href="Geometric_distribution" title="Geometric distribution">geometric</a> and <a href="Exponential_distribution" title="Exponential distribution">exponential</a> distributions are memoryless.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A <a href="Random_variable" title="Random variable">random variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is memoryless if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t)}">

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t)}</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr }">

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<annotation encoding="application/x-tex">{\displaystyle \Pr }</annotation>

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</math><img src="./50be466e52dac236652eedbbfb5660e570f2a5e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle \Pr }" loading="lazy"> is its <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> or <a href="Probability_density_function" title="Probability density function">probability density function</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete</a> or <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous</a> respectively and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> are <a href="Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a> numbers.<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-2">[2]</a> In discrete cases, the definition describes the first success in an infinite sequence of <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> <a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a>, like the number of coin flips until landing heads.<a href="#cite_note-3">[3]</a> In continuous situations, memorylessness models random phenomena, like the time between two earthquakes.<a href="#cite_note-4">[4]</a> The memorylessness property asserts that the number of previously failed trials or the elapsed time is <a href="Independence_(probability_theory)" title="Independence (probability theory)">independent</a>, or has no effect, on the future trials or lead time.

The equality <a href="Characterization_(mathematics)" title="Characterization (mathematics)">characterizes</a> the <a href="Geometric_distribution" title="Geometric distribution">geometric</a> and <a href="Exponential_distribution" title="Exponential distribution">exponential distributions</a> in discrete and continuous contexts respectively.<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-:2-5">[5]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t)}">

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<annotation encoding="application/x-tex">{\textstyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t)}</annotation>

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</math><img src="./647541ada15668211d2e6304cba137ecf7cb8983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.579ex; height:2.843ex;" alt="{\textstyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t)}" loading="lazy"> when the geometric distribution starts at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> so the equality is still satisfied.<a href="#cite_note-6">[6]</a><a href="#cite_note-:0-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Characterization_of_exponential_distribution">Characterization of exponential distribution</h2></div>

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

From the memorylessness property,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t).}">

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t).}</annotation>

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</math>The definition of <a href="Conditional_probability" title="Conditional probability">conditional probability</a> reveals that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>s)}}=\Pr(X>t).}">

<mfrac>

<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>s)}}=\Pr(X>t).}</annotation>

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</math>Rearranging the equality with the <a href="Survival_function" title="Survival function">survival function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)=\Pr(X>t)}">

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<annotation encoding="application/x-tex">{\displaystyle S(t)=\Pr(X>t)}</annotation>

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</math><img src="./97b372694d1037426d29a8260875f834e6d5de2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.468ex; height:2.843ex;" alt="{\displaystyle S(t)=\Pr(X>t)}" loading="lazy">, gives<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t+s)=S(t)S(s).}">

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<annotation encoding="application/x-tex">{\displaystyle S(t+s)=S(t)S(s).}</annotation>

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</math>This implies that for any <a href="Natural_number" title="Natural number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(kt)=S(t)^{k}.}">

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<annotation encoding="application/x-tex">{\displaystyle S(kt)=S(t)^{k}.}</annotation>

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</math>Similarly, by dividing the input of the survival function and taking the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-th root,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left({\frac {t}{k}}\right)=S(t)^{\frac {1}{k}}.}">

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<mfrac>

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<msup>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle S\left({\frac {t}{k}}\right)=S(t)^{\frac {1}{k}}.}</annotation>

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</math>In general, the equality is true for any <a href="Rational_number" title="Rational number">rational number</a> in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Since the survival function is <a href="Continuous_function" title="Continuous function">continuous</a> and rational numbers are <a href="Dense_set" title="Dense set">dense</a> in the <a href="Real_number" title="Real number">real numbers</a> (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" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)=S(1)^{t}=e^{t\ln S(1)}=e^{-\lambda t}}">

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle S(t)=S(1)^{t}=e^{t\ln S(1)}=e^{-\lambda t}}</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-\ln S(1)\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda =-\ln S(1)\geq 0}</annotation>

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</math><img src="./9a83b3566fcebfee5beff95ee3801116122d87a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.707ex; height:2.843ex;" alt="{\displaystyle \lambda =-\ln S(1)\geq 0}" loading="lazy">. This is the survival function of the exponential distribution.<a href="#cite_note-:2-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Characterization_of_geometric_distribution">Characterization of geometric distribution</h2></div>

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

From the memorylessness property,

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t).}</annotation>

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</math>

The definition of <a href="Conditional_probability" title="Conditional probability">conditional probability</a> reveals that

<mfrac>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X\geq s)}}=\Pr(X>t).}</annotation>

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From this it can be proven by induction that

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X>kt)=\Pr(X>1)^{k}.}</annotation>

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Then it follows that

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<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)=\Pr(X\leq x)=1-\Pr(X>x)=1-\Pr(X>1)^{x},}</annotation>

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and if we let

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<annotation encoding="application/x-tex">{\displaystyle p:=1-\Pr(X>1)\in [0,1]}</annotation>

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we can easily see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is geometrically distributed with some parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">; in other words

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<annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Geo} (p).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFDekkingKraaikampLopuhaäMeester2005" class="citation book cs1">Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/1-84628-168-7">A Modern Introduction to Probability and Statistics</a>. Springer Texts in Statistics. London: Springer London. p. 50. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-84628-168-7">10.1007/1-84628-168-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-85233-896-1</bdi>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFNagelSteyer2017" class="citation book cs1">Nagel, Werner; Steyer, Rolf (2017-04-04). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496">Probability and Conditional Expectation: Fundamentals for the Empirical Sciences</a>. Wiley Series in Probability and Statistics (1st ed.). Wiley. pp. 260–261. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781119243496">10.1002/9781119243496</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-119-24352-6</bdi>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFBas2019" class="citation book cs1">Bas, Esra (2019). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-3-030-32323-3">Basics of Probability and Stochastic Processes</a>. Cham: Springer International Publishing. p. 74. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-32323-3">10.1007/978-3-030-32323-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-32322-6</bdi>.</cite>

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<li id="cite_note-:2-5">^ <a href="#cite_ref-:2_5-0">a</a> <a href="#cite_ref-:2_5-1">b</a> <cite id="CITEREFRiposo2023" class="citation book cs1">Riposo, Julien (2023). <a rel="nofollow" class="external text" href="https://link.springer.com/10.1007/978-3-031-31323-3">Some Fundamentals of Mathematics of Blockchain</a>. Cham: Springer Nature Switzerland. pp. 8–9. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-31323-3">10.1007/978-3-031-31323-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-031-31322-6</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFJohnsonKempKotz2005" class="citation book cs1">Johnson, Norman L.; <a href="Adrienne_W._Kemp" title="Adrienne W. Kemp">Kemp, Adrienne W.</a>; Kotz, Samuel (2005-08-19). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/0471715816">Univariate Discrete Distributions</a>. Wiley Series in Probability and Statistics (1 ed.). Wiley. p. 210. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F0471715816">10.1002/0471715816</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-27246-5</bdi>.</cite>

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<li id="cite_note-:0-7"><a href="#cite_ref-:0_7-0">^</a> <cite id="CITEREFWeissteinRoss" class="citation web cs1">Weisstein, Eric W.; Ross, Andrew M. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Memoryless.html">"Memoryless"</a>. mathworld.wolfram.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241202153603/https://mathworld.wolfram.com/Memoryless.html">Archived</a> from the original on 2024-12-02. Retrieved 2024-07-25.</cite>

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