Characteristic function: Difference between revisions

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

* The most[[indicator common andfunction]] universal usage is asof a synonym for [[indicator functionsubset]], that is the [[Function (mathematics)|function]] <math display="block">

::<math>\mathbf{1}_A:\colon X \to \{0, 1\}</math>,

:</math> which for everya given subset ''A'' of ''X'', has value 1 at points of ''A'' and 0 at points of ''X''&nbsp;&minus;&nbsp;''A''.

* The [[Characteristic function (convex analysis)|characteristic function]] in [[convex analysis]], closely related to the indicator function of a set: <math display="block">

:*When applied to a natural number an effective procedure determines correctly if a natural number is or is not in the procedure's "set": "The '''characteristic function''' is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set" (cf Boolos-Burgess-Jeffrey (2002) p. 73).

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

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

x \not \in A.

\end{cases}</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">

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

</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 function of a [[Cooperative game theory|cooperative game]] in [[game theory]].

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

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

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

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

* The [[point characteristic function (convex analysis)]] in convex analysis:[[statistics]].

==References==

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

{{Reflist}}

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

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

* In probability theory, the [[characteristic function (probability theory)]] 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>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)\,</math>

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

* The [[characteristic polynomial]] in linear algebra

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

* The [[cooperative game]] in game theory

{{disambig}}

{{DEFAULTSORT:Characteristic Function}}

{{Set index article|mathematics}}

[[Category:Mathematical disambiguation]]

[[de:Charakteristische Funktion]]

[[it:Funzione caratteristica]]

[[pl:Funkcja charakterystyczna]]