Characteristic function

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

• The indicator function of a subset, that is the function $${\mathrm{𝟏}}_A:X\to \{0,1\},$$ which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
• The characteristic function in convex analysis, closely related to the indicator function of a set: $${\chi }_A(x):=\{\begin{array}{cc}0,& x\in A;\\ +\mathrm{\infty },& x\in ̸A.\end{array}$$
• 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: $${\phi }_X(t)=\mathrm{E}\left(e^{itX}\right),$$ where $\mathrm{E}$ 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.