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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. -->
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| {{Probability distribution|
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| name =Rademacher|
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| type =mass|
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| pdf_image =|
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| cdf_image =|
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| parameters =|
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| support =<math>k \in \{-1,1\}\,</math>|
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| pdf =<math> f(k) =
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| \begin{cases}
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| 1/2, & k = -1 \\
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| 1/2, & k = 1
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| \end{cases}
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| </math>|
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| cdf =<math> F(k) =
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| \begin{cases}
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| 0, & k < -1 \\
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| 1/2, & -1 \leq k < 1 \\
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| 1, & k \geq 1
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| \end{cases}
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| </math>|
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| mean =<math>0\,</math>|
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| median =<math>0\,</math>|
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| mode =N/A|
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| variance =<math>1\,</math>|
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| skewness =<math>0\,</math>|
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| kurtosis =<math>-2\,</math>|
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| entropy =<math>\ln(2)\,</math>|
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| mgf =<math>\cosh(t)\,</math>|
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| char =<math>\cos(t)\,</math>|
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| }}
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| 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>
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| A [[Series (mathematics)|series]] of Rademacher distributed variables can be regarded as a simple symmetrical [[random walk]] where the step size is 1.
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| ==Mathematical formulation==
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| The [[probability mass function]] of this distribution is
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| :<math> f(k) = \left\{\begin{matrix} 1/2 & \mbox {if }k=-1, \\
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| 1/2 & \mbox {if }k=+1, \\
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| 0 & \mbox {otherwise.}\end{matrix}\right.</math>
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| 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
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| :<math> f( k ) = \frac{ 1 }{ 2 } \left( \delta \left( k - 1 \right) + \delta \left( k + 1 \right) \right). </math> | |
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| ==van Zuijlen's bound==
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| 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>
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| Let ''X<sub>i</sub>'' be a set of independent Rademacher distributed random variables. Then
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| : <math> \Pr \Bigl( \Bigl | \frac{ \sum_{ i = 1 }^n X_i } { \sqrt n } \Bigr| \le 1 ) \ge 0.5. </math>
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| The bound is sharp and better than that which can be derived from the normal distribution (approximately ''Pr'' > 0.31). | |
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| ==Bounds on sums==
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| 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
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|
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| :<math> \Pr( \sum_i X_i a_i > t || a_i ||_2 ) \le e^{ - \frac{ t^2 }{ 2 } } </math>
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|
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| 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>
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| Also if ||''a''<sub>i</sub>||<sub>1</sub> is finite then
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| :<math> \Pr( \sum_i X_i a_i > t || a_i ||_1 ) = 0 </math>
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| where || ''a''<sub>i</sub> ||<sub>1</sub> is the [[Lp space|1-norm]] of the sequence { ''a''<sub>i</sub> }.
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| 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>
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| :<math> Pr( || Y || > st ) \le [ \frac{ 1 }{ c } Pr( || Y || > t ) ]^{ cs^2 } </math>
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| for some constant ''c''.
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| Let ''p'' be a positive real number. Then<ref name=Khintchine1923>Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18: 109–116</ref>
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| :<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>
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| where ''c''<sub>1</sub> and ''c''<sub>2</sub> are constants dependent only on ''p''.
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| For ''p'' ≥ 1
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| <math> c_2 \le c_1 \sqrt{ p } </math>
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| ==Applications==
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| The Rademacher distribution has been used in [[Bootstrapping (statistics)|bootstrapping]].
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| The Rademacher distribution can be used to show that [[normally distributed and uncorrelated does not imply independent]].
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| ==Related distributions==
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| * [[Bernoulli distribution]]: If ''X'' has a Rademacher distribution then <math>\frac{X+1}{2}</math> has a Bernoulli(1/2) distribution.
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| ==References==
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| {{reflist}}
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| {{ProbDistributions|discrete-finite}}
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| {{DEFAULTSORT:Rademacher Distribution}}
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| [[Category:Discrete distributions]]
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| [[Category:Probability distributions]]
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| [[it:Distribuzione discreta uniforme#Altre distribuzioni]]
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Hi, everybody!
I'm French female :).
I really like The Simpsons!
my website - FIFA Coin Generator