# Characteristic function

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

${\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}$
which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$
• In probability theory, the 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:
${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}$
where E means expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.