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

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse): Image File history File links Download high resolution version (1300x975, 125 KB) Probability density function for the Wigner semicircle distribution File links The following pages link to this file: Wigner semicircle distribution ... Image File history File links Download high resolution version (1300x975, 157 KB) Cumulative distribution function for the Wigner semicircle distribution File links The following pages link to this file: Wigner semicircle distribution ... RADIUS (Remote Authentication Dial In User Service) is an Authentication, Authorization and Accounting (AAA) protocol for applications such as network access or IP mobility. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds... In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... 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... Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A... Eugene Wigner (left) and Alvin Weinberg Eugene Paul Wigner (Hungarian Wigner Pál Jenő) (November 17, 1902 – January 1, 1995) was a Hungarian physicist and mathematician. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...

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 α (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 α (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...


References

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

External links


  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.
  More results at FactBites »

 
 

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