Characteristic function: Difference between revisions

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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''&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">

* There is an indicator function for affine varieties over a [[finite field]]:<ref>{{Cite book|title=Course in Arithmetic|last=Serre|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 = \left\{ x \in \mathbb{F}_q^n : f_\alpha(x) = 0 \right\}</math> be their vanishing locus. Then, the function <math display="inline">P(x) = \prod\left(1 - f_\alpha(x)^{q-1}\right)</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 [[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">

*: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.

*:<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]].