Characteristic function: Difference between revisions

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

Some mathematicians use the phrase '''<i>characteristic function</i>''' synonymously with "[[indicator function]]". The indicator function of a [[subset]] ''A'' of a [[set]] ''B'' is the [[function]] with ___domain ''B'', whose value is 1 at each point in ''A'' and 0 at each point that is in ''B'' but not in ''A''.

* The [[indicator function]] of a [[subset]], that is the [[Function (mathematics)|function]] <math display="block">

----

\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 display="block">

\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 number|realline]] line is given by the following formula, where ''X'' is any [[random variable]] with the distribution in question: <math display="block">

\varphi_X(t) = \operatorname{E}\left(e^{itX}\right),

</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]] in [[statistics]].

==References==

In [[probability theory]], the '''characteristic function''' of any [[probability distribution]] on the [[real number|real]] line is given by the following formula, where ''X'' is any [[random variable]] with the distribution in question:

{{Reflist}}

:<math>\varphi(t)=E\left(e^{itX}\right).</math>

Here ''t'' is a [[real number]] and E denotes the [[expected value]].

{{DEFAULTSORT:Characteristic Function}}

If ''X'' is a [[vector space|vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].

{{Set index article|mathematics}}

Related concepts include the [[moment-generating function]] and the [[probability-generating function]].