Characteristic function
In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
- The indicator function of a subset, that is the function <math display="block">
\mathbf{1}_A\colon X \to \{0, 1\}, </math> which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
- The characteristic function in convex analysis, closely related to the indicator function of a set: <math display="block">
\chi_A (x) := \begin{cases}
0, & x \in A; \\ + \infty, & x \not \in A.
\end{cases}</math>
- 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: <math display="block">
\varphi_X(t) = \operatorname{E}\left(e^{itX}\right), </math> where <math>\operatorname{E}</math> denotes 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.