Memory loss

Memorylessness is a special property of the exponential distribution and the geometric distribution . It says that the conditional probability distributions are the same for any precondition.

Memorylessness finds z. It is used, for example, in queuing theory , where - in relation to the waiting time in a queue - it means that the probability of waiting t seconds after waiting s seconds before is the same for any s . The random variable does not “remember” how long it was waited and is therefore without memory.

This fact is also used in the survival function , with which one z. For example, it is modeled that the failure probability of a component does not depend on the useful life that has already passed.

definition

The probability distribution of a random variable is memoryless if the following applies to the conditional probability: ${\ displaystyle X}$

{\ displaystyle P (X \ geq s + t \, \ mid \, X \ geq s) = {\ frac {P (X \ geq s + t)} {P (X \ geq s)}} = P ( X \ geq t)}

for everyone and . ${\ displaystyle s}$ ${\ displaystyle t}$

I.e. the conditional probability corresponds to the unconditional probability shifted by the precondition . For example: ${\ displaystyle s}$

{\ displaystyle P (X> 40 \, \ mid \, X> 30) = P (X> 10)}

Memorylessness is a defining quality. On a continuous probability space the exponential distribution is the memoryless distribution, on a discrete one it is the geometric distribution.

Exponential distribution

For the exponential distribution, inserting it into the definition gives:

{\ displaystyle P (X \ geq s + t \, \ mid \, X \ geq s) = {\ frac {P (X \ geq s + t)} {P (X \ geq s)}} = {\ frac {e ^ {- \ lambda (s + t)}} {e ^ {- \ lambda s}}} = e ^ {- \ lambda t} = P (X \ geq t)}

.

In survival time analysis , the above formula is interpreted as follows: The conditional probability that the life span exceeds the value under the condition that it has already exceeded the value is equal to the ( unconditional ) probability that the life span exceeds the value . If the service life is already units of time, the probability that the individual will survive at least further units of time is just as great as the probability that an individual of the same type will survive at least units of time. ${\ displaystyle X}$ ${\ displaystyle s + t}$ ${\ displaystyle s}$ ${\ displaystyle t}$ ${\ displaystyle s}$ ${\ displaystyle t}$ ${\ displaystyle t}$

Geometric distribution

For the geometric distribution with the definition for we get: ${\ displaystyle P (X = n) = p (1-p) ^ {n}}$ ${\ displaystyle n = 0,1,2, \ dotsc}$

{\ displaystyle {\ begin {aligned} P (X = n + k \, \ mid \, X \ geq n) & = {\ frac {P (X = n + k)} {P (X \ geq n) }} = {\ frac {p (1-p) ^ {n + k}} {\ sum _ {i = n} ^ {\ infty} p (1-p) ^ {i}}} = {\ frac {p (1-p) ^ {n + k}} {p (1-p) ^ {n} \ sum _ {i = 0} ^ {\ infty} (1-p) ^ {i}}} \ \ & = {\ frac {p (1-p) ^ {n + k}} {p (1-p) ^ {n} {\ frac {1} {1- (1-p)}}}} = {\ frac {p (1-p) ^ {n + k}} {(1-p) ^ {n}}} = p (1-p) ^ {k} \\ & = P (X = k) . \ end {aligned}}}

Markov chains

Markov chains are referred to as memoryless if the future state of the process depends only on information from the current state and not on the further past. Thus one can say that a Markov chain has a memory of length 1. This property is known as the Markov property .

literature

Christian Hesse: Applied probability theory : a well-founded introduction with over 500 realistic examples and tasks, Vieweg, Braunschweig / Wiesbaden 2003, ISBN 978-3-528-03183-1 .

Web links

[1] - Mathepedia: Exponential Distribution Section Memorylessness

Individual evidence

^ Karl Mosler and Friedrich Schmid: Probability calculation and conclusive statistics. Springer-Verlag, 2011, p. 97.

[1] Karl Mosler and Friedrich Schmid: Probability calculation and conclusive statistics. Springer-Verlag, 2011, p. 97.