Rademacher distribution

Rademacher

Support	${\displaystyle k\in \{-1,1\}\,}$
pmf	${\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.}$
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	${\displaystyle 0\,}$
Ex. kurtosis	${\displaystyle -2\,}$
Entropy	${\displaystyle \ln(2)\,}$
MGF	${\displaystyle \cosh(t)\,}$
CF	${\displaystyle \cos(t)\,}$

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 +1 and a 50% chance of being -1.

A series of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1.

The probability mass function of this distribution is

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

Van Zuijlen has proved the following result.

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

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

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

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

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

for some constant c.

Let p be a positive real number. Then

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1,

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

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