Characteristic function: Difference between revisions

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

* As a synonym for theThe [[indicator function]], that is the function

::<math>\mathbf{1}_A\colon X \to \{0, 1\},</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>\chi_{A}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>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right),</math>,

:where E means expected value. ThisFor conceptmultivariate extendsdistributions, tothe multivariateproduct distributions''tX'' is a scalar product of vectors.

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

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

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

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