In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
- The indicator function, that is the function
- which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
- There is an indicator function for affine varieties over a finite field: given a finite set of functions let be their vanishing locus. Then, the function acts as an indicator function for . If then , otherwise, for some , we have , which implies that , hence .
- The characteristic function in convex analysis, closely related to the indicator function of a set:
- In probability theory, the characteristic function 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. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
- The characteristic function of a cooperative game in game theory.
- The characteristic polynomial in linear algebra.
- The characteristic state function in statistical mechanics.
- The Euler characteristic, a topological invariant.
- The receiver operating characteristic in statistical decision theory.
- The point characteristic function in statistics.
- Serre. Course in Arithmetic. p. 5.
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