Standard error - Revision history

101.59.99.96: /* Standard error of the mean */

2014-11-30T12:14:16Z

Standard error of the mean

← Older revision		Revision as of 14:14, 30 November 2014
Line 1:		Line 1:
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109.52.143.204: The introduction of the article was too long winded, and the last part was neither clear nor in the right place. As I find extremely important and related to the subject t-student distr. I added a paragraph.

2014-03-02T08:48:17Z

The introduction of the article was too long winded, and the last part was neither clear nor in the right place. As I find extremely important and related to the subject t-student distr. I added a paragraph.

← Older revision		Revision as of 10:48, 2 March 2014
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	~~A '''causal system''' (also known as~~ a ~~[[physical system\|physical]] or '''nonanticipative system''~~'~~) is a [[system]] where the output depends on past and~~		As they call me Gabrielle. I am a particular cashier and I'm doing pretty good financially. As a girl what I totally like is going in the market to karaoke but I didn't made a dime from it. My husband and I select to reside in Guam but I will gain to move in a suitable year or two. See all that is new on my web presence here: http://circuspartypanama.com<br><br>my web site; [http://circuspartypanama.com clash of clans hack cydia]
	~~current inputs but not future inputs i~~.~~e. the output <math> y(t_{0})</math> only depends on the input <math>x(t)</math> for values of <math>t \le t_{0}</math>.~~

	~~The idea that the output of~~ a ~~function at any time depends only on past and present values of input~~ is ~~defined by~~ the ~~property commonly referred~~ to ~~as [[causality]]. A system that has~~ '~~'some'' dependence on input values~~ from the future (in addition to possible dependence on past or current input values) is termed a non-causal or [[acausal system]], and a system that depends ''solely'' on future input values is an [[anticausal system]]. ~~Note that some authors have defined an anticausal system as one that depends solely on future ''~~and ~~present'' input values or, more simply, as a system that does not depend on past input values.~~

	~~Classically, [[nature]] or physical reality has been considered~~ to ~~be a causal system. Physics involving [[special relativity]] or [[general relativity]] require more careful definitions of causality, as described elaborately~~ in ~~[[causality (physics)]].~~

	~~The causality of systems also plays an important role~~ in ~~[[digital signal processing]], where [[LTI system theory\|filters]] are constructed so that they are causal, sometimes by altering~~ a ~~non-causal formulation to remove the lack of causality so that it is realizable~~. ~~For more information, see [[causal filter]]. For a causal system, the [[impulse response]] of the system must be 0 for~~ all ~~<math>t<0</math>. That is the sole necessary as well as sufficient condition for causality of a system, linear or non-linear. Note~~ that ~~similar rules apply to either discrete or continuous cases.~~

	~~== Mathematical definitions ==~~

	~~Definition 1: A system mapping <math>x</math> to <math>y</math>~~ is ~~causal if and only if, for any pair of input signals <math>x_{1}(t)</math> and <math>x_{2}(t)</math> such that~~
	:~~<math>x_{1}(t) = x_{2}(t), \quad \forall \ t \le t_{0},</math>~~
	~~the corresponding outputs satisfy~~
	:~~<math>y_{1}(t) = y_{2}(t), \quad \forall \ t \le t_{0}.<~~/~~math>~~

	~~Definition 2: Suppose <math>h(t)<~~/~~math> is the impulse response of the system <math>H</math>~~. ~~(only fully accurate for a system described by linear constant coefficient differential equation). The system~~ <~~math~~>H<~~/math~~> ~~is causal if and only if~~
	:~~<math>h(t) = 0, \quad \forall \ t <0 </math>~~
	~~otherwise it is non-causal.~~

	~~==Examples==~~
	~~The following examples are for systems with an input <math>x<~~/~~math> and output <math>y<~~/~~math>~~.
	~~=== Examples~~ of ~~causal systems ===~~
	* Memoryless system
	~~::<math>y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)</math>~~

	* Autoregressive filter
	~~::<math>y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau</math>~~

	~~=== Examples of non-causal (acausal) systems ===~~
	*
	~~::<math>y(t)=\int_{-\infty}^\infty \sin (t+\tau) x(\tau)\,d\tau</math>~~

	* Central moving average
	~~::<math>y_n=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}</math>~~

	* For coefficients of t
	~~::<math>y \left( t \right) =x(at)</math>~~

	~~=== Examples of anti-causal systems ===~~
	*
	~~::<math>y(t) =\int _0^\infty \sin (t+\tau) x(\tau)\,d\tau</math>~~

	*Look-ahead
	~~::<math>y_n=x_{n+1}</math>~~

	~~== References ==~~
	* {{cite book \| author=Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid; with S. Hamid \| title=Signals and Systems \| publisher=Pearson Education \| year=1998 \| isbn=0-13-814757-4}}

	~~[[Category:Control theory]]~~
	~~[[Category:Digital signal processing]]~~
	~~[[Category:Systems theory]]~~
	~~[[Category:Physical systems]]~~
	~~[[Category:Dynamical systems]~~]

200.169.41.117: /* Standard error of mean versus standard deviation */

2013-12-20T19:18:00Z

Standard error of mean versus standard deviation

← Older revision		Revision as of 21:18, 20 December 2013
Line 1:		Line 1:
			A '''causal system''' (also known as a [[physical system\|physical]] or '''nonanticipative system''') is a [[system]] where the output depends on past and
			current inputs but not future inputs i.e. the output <math> y(t_{0})</math> only depends on the input <math>x(t)</math> for values of <math>t \le t_{0}</math>.

			The idea that the output of a function at any time depends only on past and present values of input is defined by the property commonly referred to as [[causality]]. A system that has ''some'' dependence on input values from the future (in addition to possible dependence on past or current input values) is termed a non-causal or [[acausal system]], and a system that depends ''solely'' on future input values is an [[anticausal system]]. Note that some authors have defined an anticausal system as one that depends solely on future ''and present'' input values or, more simply, as a system that does not depend on past input values.

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			The causality of systems also plays an important role in [[digital signal processing]], where [[LTI system theory\|filters]] are constructed so that they are causal, sometimes by altering a non-causal formulation to remove the lack of causality so that it is realizable. For more information, see [[causal filter]]. For a causal system, the [[impulse response]] of the system must be 0 for all <math>t<0</math>. That is the sole necessary as well as sufficient condition for causality of a system, linear or non-linear. Note that similar rules apply to either discrete or continuous cases.

			== Mathematical definitions ==

			Definition 1: A system mapping <math>x</math> to <math>y</math> is causal if and only if, for any pair of input signals <math>x_{1}(t)</math> and <math>x_{2}(t)</math> such that
			:<math>x_{1}(t) = x_{2}(t), \quad \forall \ t \le t_{0},</math>
			the corresponding outputs satisfy
			:<math>y_{1}(t) = y_{2}(t), \quad \forall \ t \le t_{0}.</math>

			Definition 2: Suppose <math>h(t)</math> is the impulse response of the system <math>H</math>. (only fully accurate for a system described by linear constant coefficient differential equation). The system <math>H</math> is causal if and only if
			:<math>h(t) = 0, \quad \forall \ t <0 </math>
			otherwise it is non-causal.

			==Examples==
			The following examples are for systems with an input <math>x</math> and output <math>y</math>.
			=== Examples of causal systems ===
			* Memoryless system
			::<math>y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)</math>

			* Autoregressive filter
			::<math>y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau</math>

			=== Examples of non-causal (acausal) systems ===
			*
			::<math>y(t)=\int_{-\infty}^\infty \sin (t+\tau) x(\tau)\,d\tau</math>

			* Central moving average
			::<math>y_n=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}</math>

			* For coefficients of t
			::<math>y \left( t \right) =x(at)</math>

			=== Examples of anti-causal systems ===
			*
			::<math>y(t) =\int _0^\infty \sin (t+\tau) x(\tau)\,d\tau</math>

			*Look-ahead
			::<math>y_n=x_{n+1}</math>

			== References ==
			* {{cite book \| author=Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid; with S. Hamid \| title=Signals and Systems \| publisher=Pearson Education \| year=1998 \| isbn=0-13-814757-4}}

			[[Category:Control theory]]
			[[Category:Digital signal processing]]
			[[Category:Systems theory]]
			[[Category:Physical systems]]
			[[Category:Dynamical systems]]

en>Oldmaneinstein: /* Standard error f mean versus standard deviation */ Link "descriptive statistics"

2012-08-16T22:06:35Z

Standard error f mean versus standard deviation: Link "descriptive statistics"

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