Rademacher distribution

The Rademacher distribution is a probability distribution and can therefore be assigned to the mathematical sub-area of stochastics . With her it is a simple univariate discrete probability distribution on , among other things, to define the symmetric simple random walk on is used. ${\displaystyle \{-1,1\}}$ ${\displaystyle \mathbb {Z} }$

It is named after Hans Rademacher (1892–1969).

definition

The Rademacher distribution is defined on and has the probability function ${\displaystyle \{-1,1\}}$

${\displaystyle f(n)={\begin{cases}0{,}5&{\text{ falls }}n=-1\\0{,}5&{\text{ falls }}n=1\end{cases}}}$

The distribution function is then

${\displaystyle F_{X}(t)={\begin{cases}0&{\text{ falls }}t<-1\\0{,}5&{\text{ falls }}-1\leq t<1\\1&{\text{ falls }}t\geq 1\end{cases}}}$

properties

Expected value and other measures of location

The expected value of a rademacher distributed random variable is

${\displaystyle \operatorname {E} (X)=0}$.

The median is

${\displaystyle {\tilde {m}}=0}$.

Variance

The variance corresponds to the standard deviation :

${\displaystyle \operatorname {Var} (X)=\sigma _{X}=1}$.

symmetry

The Rademacher distribution is symmetrical around the 0.

Crookedness

The crooked thing is

${\displaystyle \operatorname {v} (X)=0}$.

Excess and bulge

The excess of the Rademacher distribution is

${\displaystyle \gamma =-2}$.

That’s the bulge

${\displaystyle \beta _{2}=1}$.

Higher moments

The -th moments are ${\displaystyle k}$

${\displaystyle m_{k}={\begin{cases}0&{\text{ falls k gerade}}\\1&{\text{ falls k ungerade}}\end{cases}}}$

entropy

The entropy is

${\displaystyle \mathrm {H} (X)=\log _{2}(2)}$

measured in bits .

Accumulators

The cumulative generating function is

${\displaystyle g_{X}(t)=\ln(\cosh(t))}$.

That’s the first derivative

${\displaystyle g'_{X}(t)=\tanh(t)}$

and therefore the first cumulative . There are closed representations for the higher derivatives . ${\displaystyle \tau _{1}=0}$

Moment generating function

The moment generating function is

${\displaystyle M_{X}(t)=\cosh(t)}$.

Characteristic function

The characteristic function is

${\displaystyle \varphi _{X}(t)=\cos(t)}$.

Relationship to other distributions

Relationship to the two-point distribution

The Rademacher distribution is a two-point distribution with . ${\displaystyle a=-1,b=1,p=q=0{,}5}$

Relationship to discrete equal distribution

The Rademacher distribution is a discrete uniform distribution on . ${\displaystyle x_{1}=-1,x_{2}=1}$

Relationship to the Bernoulli distribution

Both the Bernoulli distribution with and the Rademacher distribution model a fair coin toss (or a fair, random yes / no decision). The only difference is that heads (success) and tails (failure) are coded differently. ${\displaystyle p=q=0{,}5}$

Relationship to the binomial distribution and the random walk

Are independent rademacher distributed random variables, then is ${\displaystyle X_{1},X_{2},\dots }$

${\displaystyle Y_{n}:=\sum _{i=1}^{n}X_{i}}$

exactly the symmetric random walk on . So is ${\displaystyle \mathbb {Z} }$

${\displaystyle 0{,}5\left(n+\sum _{i=1}^{n}X_{i}\right)\sim Bin(n;0{,}5)}$

so binomially distributed .

Relationship to the Laplace distribution

If rademacher is distributed and is exponentially distributed to the parameter , then laplace is distributed to the position parameter 0 and the scale parameter . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \lambda }$ ${\displaystyle X\cdot Y}$ ${\displaystyle {\frac {1}{\lambda }}}$