Characteristic function
Encyclopedia
In mathematics, characteristic function can refer to any of several distinct concepts:
- The most common and universal usage is as a synonym for indicator function, that is the function
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- which for every subset A of X, has value 1 at points of A and 0 at points of X − A.
- In probability theory, the characteristic functionCharacteristic function (probability theory)In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:
- where E means expected value. This concept extends to multivariate distributions.
- The characteristic functionCharacteristic function (convex analysis)In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership of a given element in that set...
in convex analysis:
- The characteristic state functionCharacteristic state functionThe characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble.In particular, if the partition function P satisfiesP = \exp or P = \exp...
in statistical mechanics.
- The characteristic polynomialCharacteristic polynomialIn linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
in linear algebra.
- The Euler characteristicEuler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
, a topological invariant.
- The cooperative gameCooperative gameIn game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
in game theory.