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{{For|the use of the term in [[materials science]]|hysteresis}}
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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''&nbsp;>&nbsp;''m''&nbsp;+&nbsp;''n'' | ''X''&nbsp; > &nbsp;''m'') denotes the [[conditional probability]] that the value of ''X'' is larger than ''m''&nbsp;+&nbsp;''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,&nbsp;∞). 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]]

Latest revision as of 11:09, 26 November 2014

Over time, the data on your difficult drive gets scattered. Defragmenting your difficult drive puts your information into sequential order, creating it easier for Windows to access it. As a result, the performance of the computer might enhance. An excellent registry cleaner usually allow do this task. However should you would like to defrag your PC with Windows software. Here a link to show you how.

Before actually purchasing the software it is very best to check found on the businesses that make the software. If you may discover details found on the form of standing every organization has, maybe the risk of malicious programs is reduced. Software from reputed businesses have assisted me, plus numerous other users, to create my PC run faster.. If the product description refuses to look superior to you, does not include details about the software, refuses to include the scan functions, you need to go for another 1 which ensures you're paying for what we wish.

The 'registry' is a central database that stores information, settings and options for the computer. It's actually the many widespread reason why XP runs slow and in the event you fix this issue, you might create a computer run a lot quicker. The problem is that the 'registry' stores a great deal of settings plus details about the PC... and because Windows needs to utilize numerous of these settings, any corrupted or damaged ones might directly affect the speed of your program.

The computer was pretty quickly whenever you initially bought it. Because the registry was pretty clean and free of mistakes. After time, the computer starts to run slow plus freezes up now and then. Because there are errors accumulating inside it plus some information is rewritten or even completely deleted by a incorrect uninstall of programs, improper operations, malware or different aspects. That is the reason why your computer performance decreases gradually plus become really unstable.

The tuneup utilities 2014 should come as standard with a back up plus restore center. This must be an simple to implement task.That means which when you encounter a problem with your PC after using a registry cleaning we can just restore your settings.

S/w associated error handling - If the blue screen physical memory dump occurs after the installation of s/w application or a driver it can be which there is system incompatibility. By booting into safe mode and removing the software you are able to rapidly fix this error. You might furthermore try out a "program restore" to revert to an earlier state.

Across the top of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs might show you the results of which region. The Junk Files are mainly temporary files like web information, pictures, web pages... And they are really taking up storage.

By changing the method you utilize the web we can have access more of the precious bandwidth. This might ultimately give we a quicker surfing experience. Here is a link to 3 methods to personalize your PC speed found on the net.

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