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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'' − ''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">▼
:<math>\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}}
{{DEFAULTSORT:Characteristic Function}}
▲:<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)
{{Set index article|mathematics}}
= \int_\Omega e^{itx}\, dF_X(x)
= \int_{-\infty}^{\infty} f_X(x)\, e^{itx}\,dx</math>
Here ''t'' is a [[real number]], E denotes the [[expected value]] and ''F'' is the [[cumulative distribution function]]. The last equation is only valid when ''f''--the [[probability density function]]--exists.
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]].
Characteristic function exists for any random variable.
More than that, there is a bijection between cumulative probability functions and characteristic functions.
In other words, each cumulative probability function has one and only one characteristic function that corresponds to it.
Given a characteristic function ''f'', it is possible to reconstruct the corresponding cumulative probability function:
:<math>F_X(y) - F_X(x) = \lim_{\tau \to +\infty} \frac{1} {2\pi}
\int_{-\tau}^{+\tau} \frac{e^{-itx} - e{-ity}} {it}\, \varphi_X(t)\, dt</math>
Characteristic function can also be used to find [[moment (mathematics)|moments]] of random variable. Provided that ''n''-th moment exists, ''f'' can be differentiated ''n'' times and
:<math>\operatorname{E}\left(X^n\right) = i^n\, \varphi_X^{(n)}(0)
= i^n\, \left.\frac{d^n}{dt^n}\right|_{t=0} \varphi_X(t)</math>
Related concepts include the [[moment-generating function]] and the [[probability-generating function]].
The characteristic function is closely related to the [[Fourier transform]]:
the characteristic function of a distribution with density function ''f'' is proportional to the inverse Fourier transform of ''f''.
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