Rademacher distribution

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	$1/4\,$
Skewness	$0\,$
Ex. kurtosis	$-2\,$
Entropy	$\ln(2)\,$
MGF	$\cosh(t)\,$
CF	$\cos(t)\,$

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. 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, in term of the Dirac delta function, as

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

The Rademacher distribution has been used in bootstrapping.

Related distributions

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