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In [[mathematics]], the term "'''characteristic function'''" can refer to any of several distinct concepts:
In [[set theory]] and in most areas of Mathematics the '''characteristic function''' of a [[subset]] ''A'' of a [[set]] ''X'' is the [[function (mathematics)|function]] <math>\chi_A: X \to \{0, 1\}</math> with ___domain ''X'' having value 1 at points of ''A'' and 0 at points of ''X - A''
:<math>\chi_A(x) = \begin{cases}1, &x \in A,\\0, &x \in X-A\end{cases}</math>
Sometimes* thisThe is called '''[[indicator function''', or simply '''indicator'''. An example]] of characteristica function[[subset]], that is the [[HeavisideFunction (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>\ phi_Xvarphi_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==
Elementary properties of the characteristic function are the following:
{{Reflist}}
{{DEFAULTSORT:Characteristic Function}}
<math>\chi_{\emptyset} \equiv 0\ \ \ \ \ \chi_X \equiv 1\ \ \ \ \ \chi_{A\cap B}= \chi_A \chi_B \ \ \ \ \ \chi_{A\cup B}= \chi_A + \chi_B - \chi_A \chi_B</math>
{{Set index article|mathematics}}
<math>\chi_{X -A}= 1 -\chi_A \ \ \ \ \ \chi_{A - B}= \chi_A - \chi_A \chi_B \ \ \ \ \ \chi_{A \Delta B}= \chi_A + \chi_B - 2 \chi_A \chi_B</math>
where <math>\emptyset</math> is the empty set and <math>A \Delta B = A\cup B \ - \ A\cap B</math> is the [[symmetric difference]] of <math>A</math> and <math>B</math>.
----
▲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:
▲:<math>\phi_X(t) = \operatorname{E}\left(e^{itX}\right)
= \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 distinct 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}\, \phi_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\, \phi_X^{(n)}(0)
= -i^n\, \left[\frac{d^n}{dt^n} \phi_X(t)\right]_{t=0}. </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''. In fact, the probability distribution function is equal to the Fourier transform of the characteristic function (up to a constant of proportionality and assuming the integral is defined)
:<math>f_X(x) = \frac{1} {2\pi} \int_{-\infty}^{\infty} \phi_X(t)\,e^{-itx}\, dt.</math>
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>
\phi_{S_n}(t)=\phi_{X_1}(a_1t)\phi_{X_2}(a_2t)\cdots \phi_{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]]
[[de:Charakteristische Funktion (Stochastik)]]
[[it:Funzione indicatrice]]
[[pl:Funkcja charakterystyczna]]
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