Characteristic function

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

\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.
  • 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:
\varphi_X(t) = \operatorname{E}\left(e^{itX}\right),
where E means expected value. This concept extends to multivariate distributions.
\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}