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
::$\mathbf_A\colon X \to \,$
:which for a given subset ''A'' of ''X'', has value 1 at points of ''A'' and 0 at points of ''X'' − ''A''.
* There is an indicator function for affine varieties over a finite field: given a finite set of functions f_\alpha \in \mathbb_q _1,\ldots,x_n/math> let $V = \left\$ be their vanishing locus. Then, the function $P(x) = \prod\left(1 - f_\alpha(x)^\right)$ acts as an indicator function for $V$. If $x \in V$ then $P(x) = 1$, otherwise, for some $f_\alpha$, we have $f_\alpha(x) \neq 0$, which implies that $f_\alpha(x)^ = 1$, hence $P(x) = 0$.
* The characteristic function in convex analysis, 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, 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:
*:$\varphi_X(t) = \operatorname\left(e^\right),$
*:where $\operatorname$ 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.

References

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