mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the term "characteristic function" can refer to any of several distinct concepts: * The

indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...

of a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

, that is the function

\mathbf_A\colon X \to \,

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 Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...

, closely related to the indicator function of a set:

\chi_A (x) := \begin
 0, & x \in A; \\ + \infty, &
 x \not \in A.
\end

* In

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, the characteristic function of any

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

on the

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

is given by the following formula, where ''X'' is any

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

with the distribution in question:

\varphi_X(t) = \operatorname\left(e^\right),

where

\operatorname

denotes

expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...

. For multivariate distributions, the product ''tX'' is replaced by a

scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...

of vectors. * The characteristic function of a cooperative game in

game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

. * The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

. * The characteristic state function in

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

. * The

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

, a

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

invariant. * The

receiver operating characteristic A receiver operating characteristic curve, or ROC curve, is a graph of a function, graphical plot that illustrates the performance of a binary classifier model (can be used for multi class classification as well) at varying threshold values. ROC ...

in statistical

decision theory Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...

. * The point characteristic function in

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

References

{{Set index article, mathematics