Characteristic function

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

• The most common and universal usage is as a synonym for indicator function, that is the function

:[\mathbf_A: X \to \]

which for every subset A of X, has value 1 at points of A and 0 at points of X − A.

• The characteristic state function in statistical mechanics.

• 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\left(e^\right)\,]

where "E" means expected value. See characteristic function (probability theory).

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

 

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