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Thus, a d-variate distribution is defined to be mirror symmetric when its chiral index is null. The distribution can be discrete or continuous, and the existence of a density is not required, but the inertia must be finite and non null. In the univariate case, this index was proposed as a non parametric test of symmetry. [2]
The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
This list may not reflect recent changes. List of probability distributions * Symmetric probability distribution; A. Asymptotic distribution; C. Circular distribution;
This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X 1 + X 2) / 2, where X 1, X 2 are two independent random variables with standard uniform distribution in [0, 1]. [1]
If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4 parameter distribution, which includes the normal, the uniform, the triangular, the Student t and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution.
And in fact this is a special case of a more general theorem (See p. 59 of [18]) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one).
In statistics, the normal distribution is used in classical multivariate analysis, while elliptical distributions are used in generalized multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods ...
A common special case is the symmetric Dirichlet distribution, where all of the elements making up the parameter vector have the same value. The symmetric case might be useful, for example, when a Dirichlet prior over components is called for, but there is no prior knowledge favoring one component over another.