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Support ${\displaystyle k\in \{-1,1\}\,}$ ${\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.}$ ${\displaystyle F(k)={\begin{cases}0,&k<-1\\1/2,&-1\leq k<1\\1,&k\geq 1\end{cases}}}$ ${\displaystyle 0\,}$ ${\displaystyle 0\,}$ N/A ${\displaystyle 1\,}$ ${\displaystyle 0\,}$ ${\displaystyle -2\,}$ ${\displaystyle \ln(2)\,}$ ${\displaystyle \cosh(t)\,}$ ${\displaystyle \cos(t)\,}$

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.

Mathematical formulation

The probability mass function of this distribution is

${\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.}$

In terms of the Dirac delta function, as

${\displaystyle f(k)={\frac {1}{2}}\left(\delta \left(k-1\right)+\delta \left(k+1\right)\right).}$

Van Zuijlen's bound

Van Zuijlen has proved the following result.[2]

Let Xi be a set of independent Rademacher distributed random variables. Then

${\displaystyle \Pr \left(\left|{\frac {\sum _{i=1}^{n}X_{i}}{\sqrt {n}}}\right|\leq 1\right)\geq 0.5.}$

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

Bounds on sums

Let {xi} be a set of random variables with a Rademacher distribution. Let {ai} be a sequence of real numbers. Then

${\displaystyle \Pr \left(\sum _{i}x_{i}a_{i}>t||a||_{2}\right)\leq e^{-{\frac {t^{2}}{2}}}}$

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.[3]

Let Y = Σ xiai and let Y be an almost surely convergent series in a Banach space. The for t > 0 and s ≥ 1 we have[4]

${\displaystyle \Pr \left(||Y||>st\right)\leq \left[{\frac {1}{c}}\Pr(||Y||>t)\right]^{cs^{2}}}$

for some constant c.

Let p be a positive real number. Then the Khintchine inequality says that[5]

${\displaystyle c_{1}\left[\sum {\left|a_{i}\right|^{2}}\right]^{\frac {1}{2}}\leq \left(E\left[\left|\sum {a_{i}x_{i}}\right|^{p}\right]\right)^{\frac {1}{p}}\leq c_{2}\left[\sum {\left|a_{i}\right|^{2}}\right]^{\frac {1}{2}}}$

where c1 and c2 are constants dependent only on p.

For p ≥ 1,

${\displaystyle c_{2}\leq c_{1}{\sqrt {p}}.}$

See also: Concentration inequality - a summary of tail-bounds on random variables.

Applications

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.

Random vectors with components sampled independently from the Rademacher distribution are useful for various stochastic approximations, for example:

• The Hutchinson trace estimator,[6] which can be used to efficiently approximate the trace of a matrix of which the elements are not directly accessible, but rather implicitly defined via matrix-vector products.
• SPSA, a computationally cheap, derivative-free, stochastic gradient approximation, useful for numerical optimization.

Rademacher random variables are used in the Symmetrization Inequality.

Related distributions

• Bernoulli distribution: If X has a Rademacher distribution, then ${\displaystyle {\frac {X+1}{2}}}$ has a Bernoulli(1/2) distribution.
• Laplace distribution: If X has a Rademacher distribution and Y ~ Exp(λ), then XY ~ Laplace(0, 1/λ).

References

1. ^ Hitczenko, P.; Kwapień, S. (1994). "On the Rademacher series". Probability in Banach Spaces. Progress in probability. 35. pp. 31–36. doi:10.1007/978-1-4612-0253-0_2. ISBN 978-1-4612-6682-2.
2. ^ van Zuijlen, Martien C. A. (2011). "On a conjecture concerning the sum of independent Rademacher random variables". arXiv:1112.4988. Bibcode:2011arXiv1112.4988V. Cite journal requires |journal= (help)
3. ^ Montgomery-Smith, S. J. (1990). "The distribution of Rademacher sums". Proc Amer Math Soc. 109 (2): 517–522. doi:10.1090/S0002-9939-1990-1013975-0.
4. ^ Dilworth, S. J.; Montgomery-Smith, S. J. (1993). "The distribution of vector-valued Radmacher series". Ann Probab. 21 (4): 2046–2052. arXiv:math/9206201. doi:10.1214/aop/1176989010. JSTOR 2244710. S2CID 15159626.
5. ^ Khintchine, A. (1923). "Über dyadische Brüche". Math. Z. 18 (1): 109–116. doi:10.1007/BF01192399. S2CID 119840766.
6. ^ Avron, H.; Toledo, S. (2011). "Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix". Journal of the ACM. 58 (2): 8. CiteSeerX 10.1.1.380.9436. doi:10.1145/1944345.1944349. S2CID 5827717.