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Memorylessness

From Wikipedia, the free encyclopedia
Waiting time property of certain probability distributions
For use of the term in materials science, see hysteresis. For use of the term in stochastic processes and Markov chains, see Markov property.

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

Definition

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A random variable X {\displaystyle 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)} {\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t)}where Pr {\displaystyle \Pr } {\displaystyle \Pr } is its probability mass function or probability density function when X {\displaystyle X} {\displaystyle X} is discrete or continuous respectively and m {\displaystyle m} {\displaystyle m} and n {\displaystyle n} {\displaystyle n} are nonnegative numbers.[1] [2] In discrete cases, the definition describes the first success in an infinite sequence of independent and identically distributed Bernoulli trials, 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, or has no effect, on the future trials or lead time.

The equality characterizes the geometric and exponential distributions 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)} {\textstyle \Pr(X>t+s\mid X\geq s)=\Pr(X>t)} when the geometric distribution starts at 0 {\displaystyle 0} {\displaystyle 0} instead of 1 {\displaystyle 1} {\displaystyle 1} so the equality is still satisfied.[6] [7]

Characterization of exponential distribution

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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).} {\displaystyle \Pr(X>t+s\mid X>s)=\Pr(X>t).}The definition of 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).} {\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>s)}}=\Pr(X>t).}Rearranging the equality with the survival function, S ( t ) = Pr ( X > t ) {\displaystyle 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).} {\displaystyle S(t+s)=S(t)S(s).}This implies that for any natural number k {\displaystyle k} {\displaystyle k} S ( k t ) = S ( t ) k . {\displaystyle S(kt)=S(t)^{k}.} {\displaystyle S(kt)=S(t)^{k}.}Similarly, by dividing the input of the survival function and taking the k {\displaystyle k} {\displaystyle k}-th root, S ( t k ) = S ( t ) 1 k . {\displaystyle S\left({\frac {t}{k}}\right)=S(t)^{\frac {1}{k}}.} {\displaystyle S\left({\frac {t}{k}}\right)=S(t)^{\frac {1}{k}}.}In general, the equality is true for any rational number in place of k {\displaystyle k} {\displaystyle k}. Since the survival function is continuous and rational numbers are dense in the 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, 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}} {\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} {\displaystyle \lambda =-\ln S(1)\geq 0}. This is the survival function of the exponential distribution.[5]

References

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  1. ^ a b Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). A Modern Introduction to Probability and Statistics. Springer Texts in Statistics. London: Springer London. p. 50. doi:10.1007/1-84628-168-7. ISBN 978-1-85233-896-1.
  2. ^ Pitman, Jim (1993). Probability. New York, NY: Springer New York. p. 279. doi:10.1007/978-1-4612-4374-8. ISBN 978-0-387-94594-1.
  3. ^ Nagel, Werner; Steyer, Rolf (2017年04月04日). Probability and Conditional Expectation: Fundamentals for the Empirical Sciences. Wiley Series in Probability and Statistics (1st ed.). Wiley. pp. 260–261. doi:10.1002/9781119243496. ISBN 978-1-119-24352-6.
  4. ^ Bas, Esra (2019). Basics of Probability and Stochastic Processes. Cham: Springer International Publishing. p. 74. doi:10.1007/978-3-030-32323-3. ISBN 978-3-030-32322-6.
  5. ^ a b Riposo, Julien (2023). Some Fundamentals of Mathematics of Blockchain. Cham: Springer Nature Switzerland. pp. 8–9. doi:10.1007/978-3-031-31323-3. ISBN 978-3-031-31322-6.
  6. ^ Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel (2005年08月19日). Univariate Discrete Distributions. Wiley Series in Probability and Statistics (1 ed.). Wiley. p. 210. doi:10.1002/0471715816. ISBN 978-0-471-27246-5.
  7. ^ Weisstein, Eric W.; Ross, Andrew M. "Memoryless". mathworld.wolfram.com. Archived from the original on 2024年12月02日. Retrieved 2024年07月25日.

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