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Encyclopedia > Wigner semicircle distribution
 Probability density function Cumulative distribution function Parameters radius (real) Support pdf cdf for Mean Median Mode Variance Skewness Kurtosis Entropy mgf Char. func.

for −R < x < R, and f(x) = 0 if x > R or x < − R.

This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. In probability theory and statistics, a random matrix is a matrix-valued random variable. ...

In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Free probability is a mathematical theory which studies non-commutative random variables. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...

The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution. In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ... In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...

For positive integers n, the 2nth moment of this distribution is See also moment (physics). ...

where X is any random variable with this distribution and Cn is the nth Catalan number The Catalan numbers, named after the Belgian mathematician Eugène Charles Catalan (1814—1894), form a sequence of natural numbers that occur in various counting problems in combinatorics. ...

so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)

In free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution. Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ... A partition of U into 6 blocks: a Venn diagram representation. ... In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory free probability. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...

Making the substitution x = Rcos(θ) into the defining equation for the moment generating function it can be seen that: In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...

which can be solved (see Abramowitz and Stegun §9.6.18) to yield:

where I1(z) is the modified Bessel function. Similarly, the characteristic function is given by: In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

where J1(z) is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving sin(Rtcos(θ)) is zero.) In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

In the limit of R approaching zero, the Wigner semicircle distribution becomes a Dirac delta function. The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral... Results from FactBites:

 Eugene Wigner - definition of Eugene Wigner - Labor Law Talk Dictionary (1534 words) Wigner was one of a group of renowned Jewish-Hungarian scientists and mathematicians from turn-of-the-century Budapest, including Paul Erdős, Edward Teller, John von Neumann, and Leó Szilárd. Wigner was born in Budapest, Austria-Hungary (now Hungary), into a world where middle-class people had no automobiles, radio, gas or electricity — and did not miss those things. Wigner laid the foundation for the theory of symmetries in quantum mechanics.
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