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 $${\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}$${\displaystyle \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: $${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$${\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: $${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}$${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),} where ${\displaystyle \operatorname {E} }${\displaystyle \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.

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