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${\ displaystyle \ {- 1,1 \}}$

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

The distribution function is then

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

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 ${\ displaystyle k}$

${\ displaystyle m_ {k} = {\ begin {cases} 0 & {\ text {if k is even}} \\ 1 & {\ text {if k is odd}} \ end {cases}}}$

entropy

The entropy is

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

measured in bits .

Accumulators

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

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

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

Characteristic function

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

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 ${\ displaystyle \ mathbb {Z}}$

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

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}$ ${\ displaystyle {\ frac {1} {\ lambda}}}$