Rademacher distribution

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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.

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


The Rademacher distribution is defined on and has the probability function

The distribution function is then


Expected value and other measures of location

The expected value of a rademacher distributed random variable is


The median is



The variance corresponds to the standard deviation :



The Rademacher distribution is symmetrical around the 0.


The crooked thing is


Excess and bulge

The excess of the Rademacher distribution is


That’s the bulge


Higher moments

The -th moments are


The entropy is

measured in bits .


The cumulative generating function is


That’s the first derivative

and therefore the first cumulative . There are closed representations for the higher derivatives .

Moment generating function

The moment generating function is


Characteristic function

The characteristic function is


Relationship to other distributions

Relationship to the two-point distribution

The Rademacher distribution is a two-point distribution with .

Relationship to discrete equal distribution

The Rademacher distribution is a discrete uniform distribution on .

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.

Relationship to the binomial distribution and the random walk

Are independent rademacher distributed random variables, then is

exactly the symmetric random walk on . So is

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 .