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.

[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.

The probability mass function of this distribution is In terms of the Dirac delta function, as There are various results in probability theory around analyzing the sum of i.i.d.

Let {xi} be a set of random variables with a Rademacher distribution.

Then where ||a||2 is the Euclidean norm of the sequence {ai}, t > 0 is a real number and Pr(Z) is the probability of event Z.

[2] Let Y = Σ xiai and let Y be an almost surely convergent series in a Banach space.

In 1986, Bogusław Tomaszewski proposed a question about the distribution of the sum of independent Rademacher variables.

A series of works on this question[5][6] culminated into a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.

, one gets the following bound, first shown by Van Zuijlen.

[8] The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31).

See Chapter 17 of Testing Statistical Hypotheses for example.

Random vectors with components sampled independently from the Rademacher distribution are useful for various stochastic approximations, for example: Rademacher random variables are used in the Symmetrization Inequality.