Characteristic function

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

• The indicator function, that is the function

${\mathbf {1}}_{A}\colon X\to \{0,1\},$

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:

${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$

• 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:

${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}$

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.