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
    1 A : X → { 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:
    χ A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A . {\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:
    φ X ( t ) = E ⁡ ( e i t X ) , {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}
    where 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.

References


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