Volume form - Revision history

en>Quondum: /* Riemannian volume form */ ce; linking Levi-Civita tensor

2014-11-09T17:05:27Z

Riemannian volume form: ce; linking Levi-Civita tensor

← Older revision		Revision as of 19:05, 9 November 2014
Line 1:		Line 1:
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en>Stal potaten: /* Riemannian volume form */ Added explanation of epsilon.

2014-02-23T09:22:45Z

Riemannian volume form: Added explanation of epsilon.

← Older revision		Revision as of 11:22, 23 February 2014
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	<!~~-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one. -->~~		Hi, everybody! <br>I'm French female :). <br>I really like The Simpsons!<br><br>my website - [http://tcunews.tcu.edu.tw/userinfo.php?uid=398749 FIFA Coin Generator]

	~~{{Probability distribution\|~~
	~~name =Rademacher\|~~
	~~type =mass\|~~
	~~pdf_image =\|~~
	~~cdf_image =\|~~
	~~parameters =\|~~
	~~support =<math>k \in \{-1,1\}\,</math>\|~~
	~~pdf =<math> f(k) =~~
	~~\begin{cases}~~
	~~1/2, & k = -1 \\~~
	~~1/2, & k = 1~~
	~~\end{cases}~~
	~~</math>\|~~
	~~cdf =<math> F(k) =~~
	~~\begin{cases}~~
	~~0, & k < -1 \\~~
	~~1/2, & -1 \leq k < 1 \\~~
	~~1, & k \geq 1~~
	~~\end{cases}~~
	~~</math>\|~~
	~~mean =<math>0\,</math>\|~~
	~~median =<math>0\,</math>\|~~
	~~mode =N/A\|~~
	~~variance =<math>1\,</math>\|~~
	~~skewness =<math>0\,</math>\|~~
	~~kurtosis =<math>-2\,</math>\|~~
	~~entropy =<math>\ln(2)\,</math>\|~~
	~~mgf =~~<~~math~~>~~\cosh(t)\,</math>\|~~
	~~char =<math>\cos(t)\,</math>\|~~
	}}

	~~In [[probability theory]] and [[statistics]], the~~ '''Rademacher distribution''' (which is named after [[Hans Rademacher]]) is a [[discrete probability distribution\|discrete]] [[probability distribution]] where a random variate ''X'' has a 50% chance of being either +1 or -1.<ref name=Hitczenko1994>Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36</ref>

	~~A [[Series (mathematics)\|series]] of Rademacher distributed variables can be regarded as a simple symmetrical [[random walk]] where the step size is 1.~~

	~~==Mathematical formulation==~~

	~~The [[probability mass function]] of this distribution is~~

	~~:<math> f(k) = \left\{\begin{matrix} 1/2 & \mbox {if }k=-1, \\~~
	~~1/2 & \mbox {if }k=+1, \\~~
	~~0 & \mbox {otherwise.}\end{matrix}\right.</math>~~

	It can be also written as a [[Probability density function#Link between discrete and continuous distributions\|probability density function]], in terms of the [[Dirac delta function#Applications to probability theory\|Dirac delta function]], as

	:~~<math> f( k ) = \frac{ 1 }{ 2 } \left( \delta \left( k - 1 \right) + \delta \left( k + 1 \right) \right~~). <~~/math~~>

	~~==van Zuijlen's bound==~~

	van Zuijlen has proved the following result.<ref name=vanZuijlen2011>van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. http://arxiv.org/abs/1112.4988</ref>

	~~Let ''X<sub>i</sub>'' be a set of independent Rademacher distributed random variables. Then~~

	~~: <math> \Pr \Bigl( \Bigl \| \frac{ \sum_{ i = 1 }^n X_i } { \sqrt n } \Bigr\| \le 1 ) \ge 0.5. </math>~~

	The ~~bound is sharp and better than that which can be derived from the normal distribution (approximately ''Pr'' > 0.31).~~

	~~==Bounds on sums==~~

	~~Let { ''X''~~<~~sub~~>i<~~/sub~~> ~~} be a set of random variables with a Rademacher distribution. Let { ''a''<sub>i</sub> } be a sequence of real numbers. Then~~

	~~:<math> \Pr( \sum_i X_i a_i > t \|\| a_i \|\|_2 ) \le e^{~~ - ~~\frac{ t^2 }{ 2 } } </math>~~

	~~where \|\|''a''<sub>i</sub>\|\|<sub>2</sub> is the~~ [[Euclidean norm]] of the sequence { ''a''<sub>i</sub> }, ''t'' is a real number > 0 and ''Pr''(Z) is the probability of event ''Z''.<ref name=MontgomerySmith1990>MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522</ref>

	~~Also if \|\|''a''<sub>i</sub>\|\|<sub>1</sub> is finite then~~

	~~:<math> \Pr( \sum_i X_i a_i > t \|\| a_i \|\|_1 ) = 0 </math>~~

	~~where \|\| ''a''<sub>i</sub> \|\|<sub>1</sub> is the [[Lp space\|1-norm]] of the sequence { ''a''<sub>i</sub> }.~~

	Let ''Y'' = Σ ''X''<sub>i</sub>''a''<sub>i</sub> and let ''Y'' be an almost surely convergent [[series]] in a [[Banach space]]. The for ''t'' > 0 and ''s'' ≥ 1 we have<ref name=Dilworth1993>Dilworth SJ, Montgomery-Smith SJ (1993) The distribution of vector-valued Radmacher series. Ann Probab 21 (4) 2046-2052</ref>

	~~:<math> Pr( \|\| Y \|\| > st ) \le [ \frac{ 1 }{ c } Pr( \|\| Y \|\| > t ) ]^{ cs^2 } </math>~~

	~~for some constant ''c''.~~

	~~Let ''p'' be a positive real number. Then<ref name=Khintchine1923>Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18~~: ~~109–116<~~/~~ref>~~

	~~:<math> c_1 [ \sum{ \| a_i \|^2 } ]^\frac{ 1 }{ 2 } \le ( E[ \| \sum{ a_i X_i } \|^p ] )^{ \frac{ 1 }{ p } } \le c_2 [ \sum{ \| a_i \|^2 } ]^\frac{ 1 }{ 2 } <~~/~~math>~~

	~~where ''c''<sub>1</sub> and ''c''<sub>2</sub> are constants dependent only on ''p''~~.

	~~For ''p'' ≥ 1~~

	~~<math> c_2 \le c_1 \sqrt{ p } </math>~~

	~~==Applications==~~

	~~The Rademacher distribution has been used in [[Bootstrapping (statistics)\|bootstrapping]]~~.

	~~The Rademacher distribution can be used to show that [[normally distributed and uncorrelated does not imply independent]]~~.

	~~==Related distributions==~~
	* [[Bernoulli distribution]]: If ''X'' has a Rademacher distribution then <math>\frac{X+1}{2}</~~math> has a Bernoulli(1/2) distribution~~.

	~~==References=~~=
	~~{{reflist}}~~

	~~{{ProbDistributions\|discrete-finite}}~~

	~~{{DEFAULTSORT:Rademacher Distribution}}~~
	~~[[Category:Discrete distributions]]~~
	~~[[Category:Probability distributions]]~~
	~~[[it:Distribuzione discreta uniforme#Altre distribuzioni]~~]

en>Aetheling: /* Relation to measures */ Fix style

2013-11-16T15:44:07Z

Relation to measures: Fix style

← Older revision		Revision as of 17:44, 16 November 2013
Line 1:		Line 1:
	~~Oscar~~ is ~~what my spouse enjoys~~ to ~~contact me~~ and ~~I completely dig~~ that ~~name~~. ~~Doing ceramics~~ is ~~what her family~~ and ~~her appreciate~~. ~~Minnesota has always been his house but his spouse desires them to move~~. ~~Bookkeeping~~ is ~~my occupation~~.<br><br>~~Feel free to visit my web page~~ [~~http~~://~~bankemirates~~.~~com~~/~~index~~.~~php?do~~=/~~profile-6837/info~~/ ~~bankemirates~~.~~com~~]		<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one. -->

			{{Probability distribution\|
			name =Rademacher\|
			type =mass\|
			pdf_image =\|
			cdf_image =\|
			parameters =\|
			support =<math>k \in \{-1,1\}\,</math>\|
			pdf =<math> f(k) =
			\begin{cases}
			1/2, & k = -1 \\
			1/2, & k = 1
			\end{cases}
			</math>\|
			cdf =<math> F(k) =
			\begin{cases}
			0, & k < -1 \\
			1/2, & -1 \leq k < 1 \\
			1, & k \geq 1
			\end{cases}
			</math>\|
			mean =<math>0\,</math>\|
			median =<math>0\,</math>\|
			mode =N/A\|
			variance =<math>1\,</math>\|
			skewness =<math>0\,</math>\|
			kurtosis =<math>-2\,</math>\|
			entropy =<math>\ln(2)\,</math>\|
			mgf =<math>\cosh(t)\,</math>\|
			char =<math>\cos(t)\,</math>\|
			}}

			In [[probability theory]] and [[statistics]], the '''Rademacher distribution''' (which is named after [[Hans Rademacher]]) is a [[discrete probability distribution\|discrete]] [[probability distribution]] where a random variate ''X'' has a 50% chance of being either +1 or -1.<ref name=Hitczenko1994>Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36</ref>

			A [[Series (mathematics)\|series]] of Rademacher distributed variables can be regarded as a simple symmetrical [[random walk]] where the step size is 1.

			==Mathematical formulation==

			The [[probability mass function]] of this distribution is

			:<math> f(k) = \left\{\begin{matrix} 1/2 & \mbox {if }k=-1, \\
			1/2 & \mbox {if }k=+1, \\
			0 & \mbox {otherwise.}\end{matrix}\right.</math>

			It can be also written as a [[Probability density function#Link between discrete and continuous distributions\|probability density function]], in terms of the [[Dirac delta function#Applications to probability theory\|Dirac delta function]], as

			:<math> f( k ) = \frac{ 1 }{ 2 } \left( \delta \left( k - 1 \right) + \delta \left( k + 1 \right) \right). </math>

			==van Zuijlen's bound==

			van Zuijlen has proved the following result.<ref name=vanZuijlen2011>van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. http://arxiv.org/abs/1112.4988</ref>

			Let ''X<sub>i</sub>'' be a set of independent Rademacher distributed random variables. Then

			: <math> \Pr \Bigl( \Bigl \| \frac{ \sum_{ i = 1 }^n X_i } { \sqrt n } \Bigr\| \le 1 ) \ge 0.5. </math>

			The bound is sharp and better than that which can be derived from the normal distribution (approximately ''Pr'' > 0.31).

			==Bounds on sums==

			Let { ''X''<sub>i</sub> } be a set of random variables with a Rademacher distribution. Let { ''a''<sub>i</sub> } be a sequence of real numbers. Then

			:<math> \Pr( \sum_i X_i a_i > t \|\| a_i \|\|_2 ) \le e^{ - \frac{ t^2 }{ 2 } } </math>

			where \|\|''a''<sub>i</sub>\|\|<sub>2</sub> is the [[Euclidean norm]] of the sequence { ''a''<sub>i</sub> }, ''t'' is a real number > 0 and ''Pr''(Z) is the probability of event ''Z''.<ref name=MontgomerySmith1990>MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522</ref>

			Also if \|\|''a''<sub>i</sub>\|\|<sub>1</sub> is finite then

			:<math> \Pr( \sum_i X_i a_i > t \|\| a_i \|\|_1 ) = 0 </math>

			where \|\| ''a''<sub>i</sub> \|\|<sub>1</sub> is the [[Lp space\|1-norm]] of the sequence { ''a''<sub>i</sub> }.

			Let ''Y'' = Σ ''X''<sub>i</sub>''a''<sub>i</sub> and let ''Y'' be an almost surely convergent [[series]] in a [[Banach space]]. The for ''t'' > 0 and ''s'' ≥ 1 we have<ref name=Dilworth1993>Dilworth SJ, Montgomery-Smith SJ (1993) The distribution of vector-valued Radmacher series. Ann Probab 21 (4) 2046-2052</ref>

			:<math> Pr( \|\| Y \|\| > st ) \le [ \frac{ 1 }{ c } Pr( \|\| Y \|\| > t ) ]^{ cs^2 } </math>

			for some constant ''c''.

			Let ''p'' be a positive real number. Then<ref name=Khintchine1923>Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18: 109–116</ref>

			:<math> c_1 [ \sum{ \| a_i \|^2 } ]^\frac{ 1 }{ 2 } \le ( E[ \| \sum{ a_i X_i } \|^p ] )^{ \frac{ 1 }{ p } } \le c_2 [ \sum{ \| a_i \|^2 } ]^\frac{ 1 }{ 2 } </math>

			where ''c''<sub>1</sub> and ''c''<sub>2</sub> are constants dependent only on ''p''.

			For ''p'' ≥ 1

			<math> c_2 \le c_1 \sqrt{ p } </math>

			==Applications==

			The Rademacher distribution has been used in [[Bootstrapping (statistics)\|bootstrapping]].

			The Rademacher distribution can be used to show that [[normally distributed and uncorrelated does not imply independent]].

			==Related distributions==
			* [[Bernoulli distribution]]: If ''X'' has a Rademacher distribution then <math>\frac{X+1}{2}</math> has a Bernoulli(1/2) distribution.

			==References==
			{{reflist}}

			{{ProbDistributions\|discrete-finite}}

			{{DEFAULTSORT:Rademacher Distribution}}
			[[Category:Discrete distributions]]
			[[Category:Probability distributions]]
			[[it:Distribuzione discreta uniforme#Altre distribuzioni]]

en>Quondum: /* Riemannian volume form */ Qualifyin: there is a significant semantic distinction between the matrix representation and the tensor in this context

2012-08-16T15:18:29Z

Riemannian volume form: Qualifyin: there is a significant semantic distinction between the matrix representation and the tensor in this context

New page

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