Characteristic function explained

In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

• The indicator function of a subset, that is the function $$$$

\mathbf_A\colon X \to \, 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: $$$$

\chi_A (x) := \begin 0, & x \in A; \\ + \infty, & x \not \in A.\end

• 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: $$$$

\varphi_X(t) = \operatorname\left(e^\right), where

\operatorname{E} 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.