In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
The indicator function of a subset, that is the function
which for a given subset A of X, has value 1 at points of A and 0 at points of X − A. 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:
where denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors. 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.