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Rademacher

Support	${\displaystyle k\in \{-1,1\}\,}$
PMF	${\displaystyle f(k)={\begin{cases}1/2,&k=-1\\1/2,&k=1\end{cases}}}$
CDF	${\displaystyle F(k)={\begin{cases}0,&k<-1\\1/2,&-1\leq k<1\\1,&k\geq 1\end{cases}}}$
Mean	${\displaystyle 0\,}$
Median	${\displaystyle 0\,}$
Mode	N/A
Variance	${\displaystyle 1\,}$
Skewness	${\displaystyle 0\,}$
Excess kurtosis	${\displaystyle -2\,}$
Entropy	${\displaystyle \ln(2)\,}$
MGF	${\displaystyle \cosh(t)\,}$
CF	${\displaystyle \cos(t)\,}$

In probability theory and statistics, the Rademacher distribution (named after Hans Rademacher) is a discrete probability distribution which has a 50% chance for either 1 or -1.

Mathematical formulation

The probability mass function of this distribution is

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

It can be also written as a probability density function, in terms of the Dirac delta function, as

${\displaystyle f(k)={\frac {1}{2}}\left(\delta \left(k-1\right)+\delta \left(k+1\right)\right).}$

Applications

The Rademacher distribution has been used in bootstrapping.

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

Bounds on sums

Let { X_i } be a set of random variables with a Rademacher distribution. Let { a_i } be a sequence of real numbers. Then

${\displaystyle Pr(\sum _{i}X_{i}a_{i}>t||a_{i}||_{2})\leq e^{-{\frac {t^{2}}{2}}}}$

where || a_i||₂ is the Euclidean norm of the sequence { a_i }, t is a real number > 0 and Pr(Z) is the probability of event Z.

Also if ||a_i||₁ is finite then

${\displaystyle Pr(\sum _{i}X_{i}a_{i}>t||a_{i}||_{1})=0}$

where || a_i ||₁ is the 1-norm of the sequence { a_i }.

Related distributions

• Bernoulli distribution: If X has a Rademacher distribution then ${\displaystyle {\frac {X+1}{2}}}$ has a Bernoulli(1/2) distribution.

References

1. Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36
2. MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522

• Discrete distributions