Characteristic function

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

{\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.

There is an indicator function for affine varieties over a finite field:^[1] given a finite set of functions $f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]$ let $V=\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\}$ be their vanishing locus. Then, the function $P(x)=\prod (1-f_{\alpha }(x)^{q-1})$ 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 $f_{\alpha }(x)^{q-1}=1$ , hence ${\ce {P(x)=1}}$ .
The characteristic function in convex analysis, closely related to the indicator function of a set:

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

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. For multivariate distributions, the product tX is replaced by a scalar product of vectors.

References

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