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In [[mathematics]], the term "'''characteristic function'''" can refer to any of several distinct concepts:
* The [[indicator function]] of a [[subset]], that is the [[Function (mathematics)|function]] <math display="block">
::<math>\mathbf{1}_A\colon X \to \{0, 1\},</math>
:</math> which for a given subset ''A'' of ''X'', has value 1 at points of ''A'' and 0 at points of ''X'' − ''A''.
* The [[ characteristicCharacteristic function (convex analysis)|characteristic function]] in [[convex analysis ]], closely related to the indicator function of a set: <math display="block">▼* There is an indicator function for affine varieties over a finite field:<ref>{{Cite book|title=Course in Arithmetic|last=Serre|first=|publisher=|year=|isbn=|___location=|pages=5}}</ref> given a finite set of functions <math>f_\alpha \in \mathbb{F}_q[x_1,\ldots,x_n]</math> let <math>V = \{ x \in \mathbb{F}_q^n : f_\alpha(x) = 0 \}</math> be their vanishing locus. Then, the function <math>P(x) = \prod(1 - f_\alpha(x)^{q-1})</math> acts as an indicator function for <math>V</math>. If <math>x \in V</math> then <math>P(x) = 1</math>, otherwise, for some <math>f_\alpha</math>, we have <math>f_\alpha(x) \neq 0</math>, which implies that <math>f_\alpha(x)^{q-1} = 1</math>, hence <math>P(x) = 0</math>.
\chi_A (x) := \begin{cases}
▲* The [[characteristic function (convex analysis)|characteristic function]] in convex analysis, closely related to the indicator function of a set:
::<math>\chi_A (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>
x \not \in A.
* In probability theory, the [[characteristic function (probability theory)|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: ▼\end{cases}</math>
::<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right),</math> ▼▲* In [[probability theory ]], the [[ characteristicCharacteristic function (probability theory)|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">
:where E means expected value. For multivariate distributions, the product ''tX'' is replaced by a scalar product of vectors. ▼▲::<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right), </math>
* The characteristic function of a [[cooperative game]] in game theory. ▼▲:</math> where <math>\operatorname{E }</math> meansdenotes [[expected value ]]. For [[Joint probability distribution|multivariate distributions ]], the product ''tX'' is replaced by a [[scalar product ]] of vectors.
* The [[characteristic polynomial]] in linear algebra. ▼▲* The characteristic function of a [[ Cooperative game theory|cooperative game]] in [[game theory ]].
* The [[characteristic state function]] in statistical mechanics.
* The [[Euler characteristic polynomial]], ain topological[[linear invariantalgebra]].
* The [[receivercharacteristic operatingstate characteristicfunction]] in [[statistical decision theorymechanics]].
* The [[pointEuler characteristic function]], ina [[Topology|topological]] statisticsinvariant.
* The [[receiver operating characteristic]] in statistical [[decision theory]].
▲* The [[ point characteristic polynomialfunction]] in linear algebra[[statistics]].
==References==
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