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== Gucci Logo Scarf ==


{{Probability distribution|
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  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''&nbsp;> 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]]

Latest revision as of 18:05, 9 November 2014

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