-yllion - Revision history

en>Double sharp: I could not find a ref going up to 10^8192 with the Chinese numerals. Then again, the usage of such numbers as 10^4096 is pretty much theoretical...

2015-01-13T10:36:15Z

I could not find a ref going up to 10^8192 with the Chinese numerals. Then again, the usage of such numbers as 10^4096 is pretty much theoretical...

← Older revision		Revision as of 12:36, 13 January 2015
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en>Misof: fixed a formula

2014-02-11T07:38:50Z

fixed a formula

Show changes

en>Ankit Maity: Typo fixing, typos fixed: , → , using AWB

2012-11-26T12:16:10Z

Typo fixing, typos fixed: , → , using AWB

← Older revision		Revision as of 14:16, 26 November 2012
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	~~The drop event is discharged on~~ the ~~part wherever~~ the ~~drop occurred at the top~~ of the ~~drag operation~~. A ~~perceiver would be liable for retrieving the info being dragged~~ and ~~inserting it at~~ the ~~drop location~~. ~~This event can solely hearth if a drop is desired~~. ~~it'll not hearth if~~ the ~~user off~~ the ~~drag operation~~, ~~for instance by~~ [~~http://data~~.~~gov~~.uk/~~data/search?q=pressing pressing~~] the ~~Escape key~~, ~~or if~~ the ~~push was free whereas~~ the ~~mouse wasn't over~~ a ~~sound drop target~~. ~~For data regarding this, see playacting~~ a ~~Drop~~.<br><br>~~dragend~~<br><br>~~The supply~~ of the ~~drag can receive~~ a ~~dragend event once~~ the ~~drag operation~~ is ~~complete~~, ~~whether or~~ not ~~it absolutely was roaring~~ or ~~not~~. ~~This event isn't discharged once dragging~~ a ~~file into the browser from the OS~~. ~~For~~ a ~~lot of data regarding this~~, ~~see Finishing~~ a ~~tangle~~.<br><br>~~Conclusion Drag and drop~~ is ~~one of~~ the ~~important functions~~ in ~~the online business~~.<br>~~This function is shown very clearly in the software~~: ~~"Fusisonhq 2~~.~~0 review"~~<br>~~The evaluation software~~ is ~~easy to use in~~ the ~~field of affiliate marketing~~<br><br>~~If you beloved~~ this ~~post~~ and ~~you would like to get far more data about [http~~:/~~/drapanddrop~~.~~weebly.com~~/ ~~fusionhq 2~~.0 ~~discount~~] ~~kindly stop by our own~~ [~~http:~~//~~Www~~.~~Wired~~.~~com~~/~~search?query~~=~~web~~-~~page web~~-~~page~~].		In [[probability theory]] and [[statistics]], the term '''Markov property''' refers to the memoryless property of a [[stochastic process]]. It is named after the [[Russia]]n [[mathematician]] [[Andrey Markov]].<ref>Markov, A. A. (1954). ''Theory of Algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [Jerusalem, Israel Program for Scientific Translations, 1961; available from Office of Technical Services, [[United States Department of Commerce]]] Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: ''Teoriya algorifmov''. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]</ref>

			A stochastic process has the Markov property if the [[conditional probability distribution]] of future states of the process (conditional on both past and present values) depends only upon the present state, not on the sequence of events that preceded it. A process with this property is called a ''[[Markov process]]''. The term '''strong Markov property''' is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a [[stopping time]]. Both the terms "Markov property" and "strong Markov property" have been used in connection with a particular "memoryless" property of the [[exponential distribution#Memorylessness\|exponential distribution]].<ref>[[William Feller\|Feller, W.]] (1971) ''Introduction to Probability Theory and Its Applications, Vol II'' (2nd edition),Wiley. ISBN 0-471-25709-5 (pages 9 and 20)</ref>

			The term '''Markov assumption''' is used to describe a model where the Markov property is assumed to hold, such as a [[hidden Markov model]].

			A [[Markov random field]],<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'' OUP. ISBN 0-19-850994-4</ref> extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the [[Ising model]].

			A discrete-time stochastic process satisfying the Markov property is known as a [[Markov chain]].

			==Introduction==

			A stochastic process has the Markov property if the [[conditional probability distribution]] of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be '''Markovian''' or a '''[[Markov process]]'''. The most famous Markov process is a [[Markov chain]]. [[Brownian motion]] is another well-known Markov process.

			==History==

			{{Main\|Markov chain#History}}

			==Definition==

			Let <math>(\Omega,\mathcal{F},\mathbb{P})</math> be a [[probability space]] with a [[Filtration_(mathematics)#Measure_theory\|filtration]] <math>(\mathcal{F}_s,\ s \in I)</math>, for some ([[totally ordered]]) index set <math>I</math>; and let <math>(S,\mathcal{S})</math> be a [[measurable space]]. A <math>(S,\mathcal{S})</math>-valued stochastic process <math>X=(X_t,\ t\in I)</math> adapted to the filtration is said to possess the ''Markov property'' if, for each <math>A \in \mathcal{S}</math> and each <math>s,t\in I</math> with <math>s<t</math>,

			:<math>\mathbb{P}(X_t \in A \|\mathcal{F}_s) = \mathbb{P}(X_t \in A\| X_s).</math><ref>Durrett, Rick. ''Probability: Theory and Examples''. Fourth Edition. Cambridge: Cambridge University Press, 2010.</ref>

			In the case where <math>S</math> is a discrete set with the [[Sigma-algebra#Examples\|discrete sigma algebra]] and <math>I = \mathbb{N}</math>, this can be reformulated as follows:

			:<math>\mathbb{P}(X_n=x_n\|X_{n-1}=x_{n-1} \dots X_0=x_0)=\mathbb{P}(X_n=x_n\|X_{n-1}=x_{n-1})</math>.

			==Alternative formulations==

			Alternatively, the Markov property can be formulated as follows.

			:<math>\mathbb{E}[f(X_t)\|\mathcal{F}_s]=\mathbb{E}[f(X_t)\|\sigma(X_s)]</math>

			for all <math>t\geq s\geq 0</math> and <math>f:S\rightarrow \mathbb{R}</math> bounded and measurable.

			==Strong Markov property==

			Let <math>\mathcal{F}_{\tau^+}=\{A \in \mathcal{F}: \{\tau=t\} \cap A \in \mathcal{F}_{t+} ,\, \forall t \geq 0\}</math>.

			Suppose that <math>X=(X_t:t\geq 0)</math> is a [[stochastic process]] on a [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math> with natural [[filtration (mathematics)\|filtration]] <math>\{\mathcal{F}_t\}_{t\geq 0}</math>. Then <math>X</math> is said to have the strong Markov property if, for each [[stopping time]] <math>\tau</math>, conditioned on the event <math>\{\tau < \infty\}</math>, we have that for each <math>t\ge 0</math>, <math>X_{\tau + t}</math> is independent of <math>\mathcal{F}_{\tau^+}</math> given <math>X_\tau</math>.

			The strong Markov property implies the ordinary Markov property, since by taking the stopping time <math>\tau=t</math>, the ordinary Markov property can be deduced.

			<!-- Hide for now ... name '''Markov-type property''' seems doubtful
			==Markov Type Property==
			A stochastic process has a '''Markov-type property''' if the process's [[random variable]]s determine a set of probabilities can be factored in a way that yields the '''Markov property'''. Useful in applied research, members of such classes{{Clarify\|October 2009\|date=October 2009}} defined by their mathematics or area of application{{Clarify\|October 2009\|date=October 2009}} are referred to as '''[[Markov random field]]s'''., and occur in many situations. The [[Ising model]] is a prototypical example. -->

			==Applications==

			A very important{{Citation needed\|date=March 2012}} application of the Markov property in a generalized form is in [[Markov chain Monte Carlo]] computations in the context of [[Bayesian statistics]].

			==See also==
			*[[Markov chain]]
			*[[Markov blanket]]
			*[[Markov decision process]]
			*[[Causal Markov condition]]
			*[[Markov model]]
			*[[Chapman–Kolmogorov equation]]

			== References ==

			{{Reflist}}

			[[Category:Markov models]]
			[[Category:Markov processes]]

71.252.26.204 at 04:26, 8 July 2012

2012-07-08T04:26:19Z

New page

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