# Memorylessness

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Waiting time property of certain probability distributions

For use of the term in [materials science](/source/Materials_science), see [hysteresis](/source/Hysteresis). For use of the term in [stochastic processes](/source/Stochastic_process) and [Markov chains](/source/Markov_chain), see [Markov property](/source/Markov_property).

In [probability](/source/Probability) and [statistics](/source/Statistics), **memorylessness** is a property of [probability distributions](/source/Probability_distribution). It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the [geometric](/source/Geometric_distribution) and [exponential](/source/Exponential_distribution) distributions are memoryless.

## Definition

A [random variable](/source/Random_variable) X {\displaystyle X} is memoryless if Pr ( X > t + s ∣ X > s ) = Pr ( X > t ) {\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t)} where Pr {\displaystyle \Pr } is its [probability mass function](/source/Probability_mass_function) or [probability density function](/source/Probability_density_function) when X {\displaystyle X} is [discrete](/source/Discrete_random_variable) or [continuous](/source/Continuous_random_variable) respectively and t {\displaystyle t} and s {\displaystyle s} are [nonnegative](/source/Nonnegative) numbers.[1][2] In discrete cases, the definition describes the first success in an infinite sequence of [independent and identically distributed](/source/Independent_and_identically_distributed_random_variables) [Bernoulli trials](/source/Bernoulli_trial), like the number of coin flips until landing heads.[3] In continuous situations, memorylessness models random phenomena, like the time between two earthquakes.[4] The memorylessness property asserts that the number of previously failed trials or the elapsed time is [independent](/source/Independence_(probability_theory)), or has no effect, on the future trials or lead time.

The equality [characterizes](/source/Characterization_(mathematics)) the [geometric](/source/Geometric_distribution) and [exponential distributions](/source/Exponential_distribution) in discrete and continuous contexts respectively.[1][5] 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 Pr ( X > t + s ∣ X ≥ s ) = Pr ( X > t ) {\textstyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t)} when the geometric distribution starts at 0 {\displaystyle 0} instead of 1 {\displaystyle 1} so the equality is still satisfied.[6][7]

## Characterization of exponential distribution

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

From the memorylessness property, Pr ( X > t + s ∣ X > s ) = Pr ( X > t ) . {\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t).} The definition of [conditional probability](/source/Conditional_probability) reveals that Pr ( X > t + s ) Pr ( X > s ) = Pr ( X > t ) . {\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>s)}}=\Pr(X>t).} Rearranging the equality with the [survival function](/source/Survival_function), S ( t ) = Pr ( X > t ) {\displaystyle S(t)=\Pr(X>t)} , gives S ( t + s ) = S ( t ) S ( s ) . {\displaystyle S(t+s)=S(t)S(s).} This implies that for any [natural number](/source/Natural_number) k {\displaystyle k} S ( k t ) = S ( t ) k . {\displaystyle S(kt)=S(t)^{k}.} Similarly, by dividing the input of the survival function and taking the k {\displaystyle k} -th root, S ( t k ) = S ( t ) 1 k . {\displaystyle S\left({\frac {t}{k}}\right)=S(t)^{\frac {1}{k}}.} In general, the equality is true for any [rational number](/source/Rational_number) in place of k {\displaystyle k} . Since the survival function is [continuous](/source/Continuous_function) and rational numbers are [dense](/source/Dense_set) in the [real numbers](/source/Real_number) (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, S ( t ) = S ( 1 ) t = e t ln ⁡ S ( 1 ) = e − λ t {\displaystyle S(t)=S(1)^{t}=e^{t\ln S(1)}=e^{-\lambda t}} where λ = − ln ⁡ S ( 1 ) ≥ 0 {\displaystyle \lambda =-\ln S(1)\geq 0} . This is the survival function of the exponential distribution.[5]

## Characterization of geometric distribution

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

From the memorylessness property, Pr ( X > t + s ∣ X ≥ s ) = Pr ( X > t ) . {\displaystyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t).} The definition of [conditional probability](/source/Conditional_probability) reveals that Pr ( X > t + s ) Pr ( X ≥ s ) = Pr ( X > t ) . {\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X\geq s)}}=\Pr(X>t).} From this it can be proven by induction that Pr ( X > k ) = Pr ( X > 1 ) k . {\displaystyle \Pr(X>k)=\Pr(X>1)^{k}.} Then it follows that f X ( x ) = Pr ( X ≤ x ) = 1 − Pr ( X > x ) = 1 − Pr ( X > 1 ) x , {\displaystyle f_{X}(x)=\Pr(X\leq x)=1-\Pr(X>x)=1-\Pr(X>1)^{x},} and if we let p := 1 − Pr ( X > 1 ) ∈ [ 0 , 1 ] {\displaystyle p:=1-\Pr(X>1)\in [0,1]} we can easily see that X {\displaystyle X} is geometrically distributed with some parameter p {\displaystyle p} ; in other words X ∼ Geo ⁡ ( p ) . {\displaystyle X\sim \operatorname {Geo} (p).}

## References

1. ^ [***a***](#cite_ref-:1_1-0) [***b***](#cite_ref-:1_1-1) Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). [*A Modern Introduction to Probability and Statistics*](http://link.springer.com/10.1007/1-84628-168-7). Springer Texts in Statistics. London: Springer London. p. 50. [doi](/source/Doi_(identifier)):[10.1007/1-84628-168-7](https://doi.org/10.1007%2F1-84628-168-7). [ISBN](/source/ISBN_(identifier)) [978-1-85233-896-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-896-1).

1. **[^](#cite_ref-2)** Pitman, Jim (1993). [*Probability*](http://link.springer.com/10.1007/978-1-4612-4374-8). New York, NY: Springer New York. p. 279. [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-4374-8](https://doi.org/10.1007%2F978-1-4612-4374-8). [ISBN](/source/ISBN_(identifier)) [978-0-387-94594-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-94594-1).

1. **[^](#cite_ref-3)** Nagel, Werner; Steyer, Rolf (2017-04-04). [*Probability and Conditional Expectation: Fundamentals for the Empirical Sciences*](https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496). Wiley Series in Probability and Statistics (1st ed.). Wiley. pp. 260–261. [doi](/source/Doi_(identifier)):[10.1002/9781119243496](https://doi.org/10.1002%2F9781119243496). [ISBN](/source/ISBN_(identifier)) [978-1-119-24352-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-24352-6).

1. **[^](#cite_ref-4)** Bas, Esra (2019). [*Basics of Probability and Stochastic Processes*](http://link.springer.com/10.1007/978-3-030-32323-3). Cham: Springer International Publishing. p. 74. [doi](/source/Doi_(identifier)):[10.1007/978-3-030-32323-3](https://doi.org/10.1007%2F978-3-030-32323-3). [ISBN](/source/ISBN_(identifier)) [978-3-030-32322-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-32322-6).

1. ^ [***a***](#cite_ref-:2_5-0) [***b***](#cite_ref-:2_5-1) Riposo, Julien (2023). [*Some Fundamentals of Mathematics of Blockchain*](https://link.springer.com/10.1007/978-3-031-31323-3). Cham: Springer Nature Switzerland. pp. 8–9. [doi](/source/Doi_(identifier)):[10.1007/978-3-031-31323-3](https://doi.org/10.1007%2F978-3-031-31323-3). [ISBN](/source/ISBN_(identifier)) [978-3-031-31322-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-31322-6).

1. **[^](#cite_ref-6)** Johnson, Norman L.; [Kemp, Adrienne W.](/source/Adrienne_W._Kemp); Kotz, Samuel (2005-08-19). [*Univariate Discrete Distributions*](https://onlinelibrary.wiley.com/doi/book/10.1002/0471715816). Wiley Series in Probability and Statistics (1 ed.). Wiley. p. 210. [doi](/source/Doi_(identifier)):[10.1002/0471715816](https://doi.org/10.1002%2F0471715816). [ISBN](/source/ISBN_(identifier)) [978-0-471-27246-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-27246-5).

1. **[^](#cite_ref-:0_7-0)** Weisstein, Eric W.; Ross, Andrew M. ["Memoryless"](https://mathworld.wolfram.com/Memoryless.html). *mathworld.wolfram.com*. [Archived](https://web.archive.org/web/20241202153603/https://mathworld.wolfram.com/Memoryless.html) from the original on 2024-12-02. Retrieved 2024-07-25.

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