Rademacher distribution

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

In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate X has a 50% chance of being either +1 or -1.^[1]

A 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

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

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

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

van Zuijlen's bound

van Zuijlen has proved the following result.^[2]

Let X_i be a set of independent Rademacher distributed random variables. Then

$\Pr {\Bigl (}{\Bigl |}{\frac {\sum _{{i=1}}^{n}X_{i}}{{\sqrt n}}}{\Bigr |}\leq 1)\geq 0.5.$

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

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

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.^[3]

Also if ||a_i||₁ is finite then

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

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

Let Y = Σ X_ia_i and let Y be an almost surely convergent series in a Banach space. The for t > 0 and s ≥ 1 we have^[4]

$Pr(||Y||>st)\leq [{\frac {1}{c}}Pr(||Y||>t)]^{{cs^{2}}}$

for some constant c.

Let p be a positive real number. Then^[5]

$c_{1}[\sum {|a_{i}|^{2}}]^{{\frac {1}{2}}}\leq (E[|\sum {a_{i}X_{i}}|^{p}])^{{{\frac {1}{p}}}}\leq c_{2}[\sum {|a_{i}|^{2}}]^{{\frac {1}{2}}}$

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1

$c_{2}\leq c_{1}{\sqrt {p}}$

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.

Related distributions

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

References

↑ Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36
↑ van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. http://arxiv.org/abs/1112.4988
↑ MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522
↑ Dilworth SJ, Montgomery-Smith SJ (1993) The distribution of vector-valued Radmacher series. Ann Probab 21 (4) 2046-2052
↑ Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18: 109–116

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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