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If a symmetric distribution is unimodal, the mode coincides with the median and mean. All odd central moments of a symmetric distribution equal zero (if they exist), because in the calculation of such moments the negative terms arising from negative deviations from x 0 {\displaystyle x_{0}} exactly balance the positive terms arising from equal ...
Additional families of symmetric graphs with an even number of vertices 2n, are the evenly split complete bipartite graphs K n,n and the crown graphs on 2n vertices. Many other symmetric graphs can be classified as circulant graphs (but not all). The Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree.
Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A = (a ij), then a ij = a ji, for all indices i and j. For example, the following 3×3 matrix is symmetric:
Benford's law, which describes the frequency of the first digit of many naturally occurring data. The ideal and robust soliton distributions. Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory. In statistics , symmetry also manifests as symmetric probability distributions , and as skewness —the asymmetry of distributions.
In statistics, an -sample statistic (a function in variables) that is obtained by bootstrapping symmetrization of a -sample statistic, yielding a symmetric function in variables, is called a U-statistic. Examples include the sample mean and sample variance.
A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if: [1], (), where the notation aRb means that (a, b) ∈ R. An example is the relation "is equal to", because if a = b is true then b = a is also true.
It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led Benoît Mandelbrot to propose that cotton prices follow an alpha-stable distribution ...