Characteristic function: Difference between revisions

Revision as of 23:16, 4 December 2015

Latest revision as of 18:49, 6 March 2024

(19 intermediate revisions by 18 users not shown)

Line 1:

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

* The As[[indicator afunction]] synonymof for thea [[indicator functionsubset]], 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''&nbsp;&minus;&nbsp;''A''.

* In probability theory, theThe [[characteristicCharacteristic function (probabilityconvex theoryanalysis)|characteristic function]] ofin any[[convex probabilityanalysis]], distributionclosely onrelated the real line is given byto the followingindicator formula,function whereof ''X''a isset: any<math random variable with the distribution in question:display="block">

\chi_A (x) := \begin{cases}

::<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)</math>,

0, & x \in A; \\ + \infty, &

:where E means expected value. This concept extends to multivariate distributions.

x \not \in A.

* The [[characteristic function (convex analysis) | characteristic function]] in convex analysis:

\end{cases}</math>

::<math>\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>

* 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: <math display="block">

* The characteristic function of a [[cooperative game]] in game theory.

::<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)</math>,

* The [[characteristic polynomial]] in linear algebra.

</math> where <math>\operatorname{E}</math> denotes [[expected value]]. For [[Joint probability distribution|multivariate distributions]], the product ''tX'' is replaced by a [[scalar product]] of vectors.

* The [[characteristic state function]] in statistical mechanics.

* The characteristic function of a [[Cooperative game theory|cooperative game]] in [[game theory]].

* The [[Euler characteristic]], a topological invariant.

* The [[Receiver operating characteristic polynomial]] in statistical[[linear decision theoryalgebra]].

* The [[point characteristic state function]] in statistics[[statistical mechanics]].

* The [[Euler characteristic]], a [[Topology|topological]] invariant.

* The [[receiver operating characteristic]] in statistical [[decision theory]].

* The [[point characteristic polynomialfunction]] in linear algebra[[statistics]].

==References==

{{Reflist}}

{{disambig}}

{{DEFAULTSORT:Characteristic Function}}

{{Set index article|mathematics}}

[[Category:Mathematical disambiguation]]