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{{For|the use of the term in [[materials science]]|hysteresis}}

In [[probability]] and [[statistics]], '''memorylessness''' is a property of certain [[probability distribution]]s: the [[exponential distribution]]s of non-negative real numbers and the [[geometric distribution]]s of non-negative integers.

The property is most easily explained in terms of "waiting times". Suppose that a [[random variable]], ''X'', is defined to be the time elapsed in a shop from 9 am on a certain day until the arrival of the first customer: thus ''X'' is the time a server ''waits'' for the first customer. The "memoryless" property makes a comparison between the probability distributions of the time a server has to wait from 9 am onwards for his first customer, and the time that the server still has to wait for the first customer on those occasions when no customer has arrived by any given later time: the property of memorylessness is that these distributions of "time from now to the next customer" are exactly the same.

The terms "memoryless" and "memorylessness" are used in a very different way to refer to [[Markov process]]es in which the underlying assumption of the [[Markov property]] implies that the properties of random variables related to the future depend only on relevant information about the current time, not on information from further in the past. The present article describes the use outside the Markov property, limited to conditional probability distributions.

==Discrete memorylessness==
Suppose ''X'' is a [[discrete random variable|discrete]] [[random variable]] whose values lie in the set { 0, 1, 2, ... }.
The probability distribution of ''X'' is '''memoryless''' precisely if for any ''m'', ''n'' in { 0, 1, 2, ... }, we have

:<math>\Pr(X>m+n \mid X > m)=\Pr(X>n).</math>

Here, Pr(''X'' > ''m'' + ''n'' | ''X''  >  ''m'') denotes the [[conditional probability]] that the value of ''X'' is larger than ''m'' + ''n'', given that it is larger than or equal to ''m''.

The ''only'' memoryless discrete probability distributions are the [[geometric distribution]]s, which feature the number of [[statistical independence|independent]] [[Bernoulli trial]]s needed to get one "success", with a fixed probability ''p'' of "success" on each trial. In other words those are the distributions of [[Negative binomial distribution#Waiting time in a Bernoulli process|waiting time in a Bernoulli process]].

===A frequent misunderstanding===
"Memorylessness" of the probability distribution of the number of trials ''X'' until the first success means that

:<math>\mathrm{}\ \Pr(X>40 \mid X > 30)=\Pr(X>10).\,</math>

It does ''not'' mean that

:<math>\mathrm{}\ \Pr(X>40 \mid X > 30)=\Pr(X>40)\,</math>

which would be true only if the events ''X'' > 40 and ''X'' > 30 were [[statistical independence|independent]].
However, we can make some valid rearrangements,

:<math> \Pr(X>40 \mid X > 30) = \frac{ \Pr(X>40 , X > 30) }{ \Pr(X > 30) }= \frac{ \Pr(X>40) }{ \Pr(X > 30) } </math> .

==Continuous memorylessness==
Suppose ''X'' is a continuous random variable whose values lie in the non-negative real numbers [0, ∞). The probability distribution of ''X'' is '''memoryless''' precisely if for any non-negative [[real number]]s ''t'' and ''s'', we have

:<math>\Pr(X>t+s \mid X>t)=\Pr(X>s).\,</math>

This is similar to the discrete version except that ''s'' and ''t'' are constrained only to be non-negative real numbers instead of [[integer]]s. Rather than counting trials until the first "success", for example, we may be marking time until the arrival of the first phone call at a switchboard.

[[Geometric distribution]]s and [[exponential distribution]]s are discrete and continuous analogs.

===The memoryless distributions are the exponential distributions===
The only memoryless continuous probability distributions are the [[exponential distribution]]s, so memorylessness completely [[characterization (mathematics)|characterizes]] the exponential distributions among all continuous ones.

To see this, first define the [[survival function]], ''G'', as

:<math>G(t) = \Pr(X > t).\,</math>

Note that ''G''(''t'') is then [[monotonically decreasing]]. From the relation

:<math>\Pr(X > t + s | X > t) = \Pr(X > s)\,</math>

and the definition of [[conditional probability]], it follows that

:<math>{\Pr(X > t + s) \over \Pr(X > t)} = \Pr(X > s).</math>

This gives the [[functional equation]]

:<math>G(t + s) = G(t) G(s)\,</math>

and solutions of this can be sought under the condition that ''G'' is a [[Monotonic function|monotone decreasing function]] (meaning that for times <math>x\leq y</math>, then <math>G(x)\geq G(y)</math>).

The functional equation alone will imply that ''G'' restricted to [[rational number|rational]] multiples of any particular number is an [[exponential function]]. Combined with the fact that ''G'' is monotone, this implies that ''G'' over its whole domain is an exponential function.

{{more footnotes|date=June 2011}}

==Notes==
{{reflist}}

==References==
* [[William Feller|Feller, W.]] (1971) ''Introduction to Probability Theory and Its Applications, Vol II'' (2nd edition),Wiley. Section I.3 ISBN 0-471-25709-5

[[Category:Theory of probability distributions]]
[[Category:Characterization of probability distributions]]

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