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In [[mathematics]], the term "'''characteristic function'''" can refer to any of several distinct concepts:
Some mathematicians use the phrase '''''characteristic function''''' synonymously with ''[[indicator function]]''. The indicator function of a [[subset]] ''A'' of a [[set]] ''B'' is the [[function (mathematics)|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]].
analog* forThe 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 form is valid only when the [[probability density function]] ''f'' exists. The form preceding it is a [[Riemann-Stieltjes integral]] and is valid regardless of whether a 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]].
A characteristic function exists for any random variable. More than that, there is a [[bijection]] between cumulative probability distribution functions and characteristic functions. In other words, two probability distributions never share the same characteristic function.
Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function ''F'':
:<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>
In general this is an [[improper integral]]; the function being integrated may be only conditionally integrable rather than [[Lebesgue integral|Lebesgue-integrable]], i.e. the integral of its absolute value may be infinite.
Characteristic functions are used in the most frequently seen proof of the [[central limit theorem]].
Characteristic functions can also be used to find [[moment (mathematics)|moments]] of random variable. Provided that ''n''-th moment exists, characteristic function 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''.
Characteristic functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example, if ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>n</sub> is a sequence of independent (and not necessarily identically distributed) random variables, and
::<math>S_n = \sum_{i=1}^n a_i X_i,</math>
:where the ''a'' <sub>i</sub> are constants, then the characteristic function for ''S'' <sub>n</sub> is given by
::<math>
\varphi_{S_n}(t)=\varphi_{X_1}(a_1t)\varphi_{X_2}(a_2t)\cdots \varphi_{X_n}(a_nt).
</math>
<!------------------------
Below was lifted from [[generating function]] ... there should be an
▲analog for the characteristic function
*Suppose that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function ''G''<sub>''N''</sub>. If the ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>N</sub> are independent <i>and</i> identically distributed with common probability-generating function ''G''<sub>X</sub>, then
::<math>G_{S_N}(z) = G_N(G_X(z)).</math>
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[[Category:probability theory]]
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