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">

*:<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">

*:<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]].