Memorylessness Explained

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

is memoryless if $$\backslash Pr(X>t+s\; \backslash mid\; X>s)=\backslash Pr(X>t)$$where is its probability mass function or probability density function when is discrete or continuous respectively and and 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.^[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 $\backslash Pr(X>t+s\; \backslash mid\; X\; \backslash geq\; s)=\backslash Pr(X>t)$ when the geometric distribution starts at

instead of 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,$$\backslash Pr(X>t+s\; \backslash mid\; X>s)=\backslash Pr(X>t).$$The definition of conditional probability reveals that$$\backslash frac\; =\; \backslash Pr(X\; >\; t).$$Rearranging the equality with the survival function,

, gives$$S(t\; +\; s)\; =\; S(t)\; S(s).$$This implies that for any natural number $$S(kt)\; =\; S(t)^k.$$Similarly, by dividing the input of the survival function and taking the -th root,$$S\backslash left(\backslash frac\backslash right)\; =\; S(t)^.$$In general, the equality is true for any rational number in place of . 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,$$S(t)\; =\; S(1)^t\; =\; e^\; =\; e^$$where . This is the survival function of the exponential distribution.

Characterization of geometric distribution

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

From the memorylessness property,$$\backslash Pr(X>t+s\; \backslash mid\; X\backslash geq\; s)=\backslash Pr(X>t).$$The definition of conditional probability reveals that$$\backslash frac\; =\; \backslash Pr(X\; >\; t).$$From this it can be proven by induction that$$\backslash Pr(X\; >\; k)\; =\; \backslash Pr(X\; >\; 1)^k.$$Then it follows that$$f\_X(x)=\backslash Pr(X\backslash leq\; x)=1-\backslash Pr(X>x)=1-\backslash Pr(X>1)^x,$$and if we let$$p\; :=\; 1\; -\; \backslash Pr(X>1)\; \backslash in\; [0,\; 1]$$we can easily see that

is geometrically distributed with some parameter ; in other words$$X\backslash sim\; \backslash operatorname(p).$$

Notes and References

1. Book: Dekking, Frederik Michel . A Modern Introduction to Probability and Statistics . Kraaikamp . Cornelis . Lopuhaä . Hendrik Paul . Meester . Ludolf Erwin . Springer London . 2005 . 978-1-85233-896-1 . Springer Texts in Statistics . London . 50 . 10.1007/1-84628-168-7.
2. Book: Pitman, Jim . Probability . Springer New York . 1993 . 978-0-387-94594-1 . New York, NY . 279 . en . 10.1007/978-1-4612-4374-8.
3. Book: Nagel, Werner . Probability and Conditional Expectation: Fundamentals for the Empirical Sciences . Steyer . Rolf . 2017-04-04 . Wiley . 978-1-119-24352-6 . 1st . Wiley Series in Probability and Statistics . 260–261 . en . 10.1002/9781119243496.
4. Book: Bas, Esra . Basics of Probability and Stochastic Processes . Springer International Publishing . 2019 . 978-3-030-32322-6 . Cham . 74 . en . 10.1007/978-3-030-32323-3.
5. Book: Riposo, Julien . Some Fundamentals of Mathematics of Blockchain . Springer Nature Switzerland . 2023 . 978-3-031-31322-6 . Cham . 8–9 . en . 10.1007/978-3-031-31323-3.
6. Book: Johnson, Norman L. . Univariate Discrete Distributions . Kemp . Adrienne W.. Adrienne W. Kemp . Kotz . Samuel . 2005-08-19 . Wiley . 978-0-471-27246-5 . 1 . Wiley Series in Probability and Statistics . 210 . en . 10.1002/0471715816.
7. Web site: Weisstein . Eric W. . Ross . Andrew M. . Memoryless . live . https://web.archive.org/web/20241202153603/https://mathworld.wolfram.com/Memoryless.html . 2024-12-02 . 2024-07-25 . mathworld.wolfram.com . en.