Characteristic function

Characteristic function

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{1}_A: X \to \{0, 1\}
which for every subset A of X, has value 1 at points of A and 0 at points of X − A.
  • In probability theory, the characteristic function (probability theory) 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}

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  • characteristic function — noun A function which is equal to 1 for all points in its domain which belong to a given set, and is equal to 0 for all points in the domain which do not belong to that given set. The characteristic function of a set A is often represented as… …   Wiktionary

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  • Characteristic function (probability theory) — The characteristic function of a uniform U(–1,1) random variable. This function is real valued because it corresponds to a random variable that is symmetric around the origin; however in general case characteristic functions may be complex valued …   Wikipedia

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  • characteristic function — Math. 1. a function defined on a given set, having value 1 for every element of the set and value 0 for every element not contained in the set. 2. See characteristic polynomial. [1875 80] * * * …   Universalium

  • characteristic function — noun : eigenfunction …   Useful english dictionary

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  • Characteristic state function — The characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble. In particular, if the partition function P satisfies P = exp( − βQ) or P = exp( + βQ) in which Q is a… …   Wikipedia

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