Characteristic function

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

\mathbf{1}_A\colon X \to \{0, 1\}, $$ which for a given subset A of X, has value 1 at points of A and 0 at points of X &minus; A. \chi_A (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}$$ \varphi_X(t) = \operatorname{E}\left(e^{itX}\right), $$ where $$\operatorname{E}$$ denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
 * The indicator function of a subset, that is the function $$
 * The characteristic function in convex analysis, closely related to the indicator function of a set: $$
 * In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: $$
 * The characteristic function of a cooperative game in game theory.
 * The characteristic polynomial in linear algebra.
 * The characteristic state function in statistical mechanics.
 * The Euler characteristic, a topological invariant.
 * The receiver operating characteristic in statistical decision theory.
 * The point characteristic function in statistics.