Characteristic function

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

As a synonym for indicator function, that is the function

\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.

The characteristic function in convex analysis:

\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}

The 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.