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Rademacher distribution
Abstract:-
The Rademacher distribution is a recoding of the Bernoulli distribution with two possible values {-1, 1}.
It's second moment (the variance) equals 1; all other moments equal 0 . It is named after German-American
mathematician Hans Rademacher and denoted Rad½.
Like the Bernoulli, a random variable has a 50% chance of a success and 50% chance of failure.
Bernoulli: 0 (failure) and 1 (success),
Rademacher: -1 (failure) and 1 (success).
The distribution is used for formulating statistical proofs, random sampling , and bootstrapping ,
where weights dg = {−1, 1} are called Rademacher weights
The Rademacher distribution has been used in bootstrapping. The Rademacher distribution can be used
to show that normally distributed and uncorrelated does not imply independent.also used to efficiently approximate
the trace of a matrix
Introduction:-
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 (that is, a sum) of Rademacher distributed variables can be regarded as a simple symmetrical random walk
where the step size is 1.
Formulas of the distribution:-
The probability mass function of this distribution is
In terms of the Dirac delta function, as
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Bounds on sums of independent Rademacher variables:
There are various results in probability theory around analyzing the sum of i.i.d. Rademacher variables,
including concentration inequalities such as
Bernstein inequalities as well as anti-concentration inequalities like Tomaszewski’s conjecture.
Concentration inequalities: