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





1


A


:
X
→
{
0
,
1
}
,


{\displaystyle \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: given a finite set of functions




f

α


∈


F


q


[

x

1


,
…
,

x

n


]


{\displaystyle f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]}
let



V
=

{

x
∈


F


q


n


:

f

α


(
x
)
=
0

}



{\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}}
be their vanishing locus. Then, the function



P
(
x
)
=
∏

(

1
−

f

α


(
x

)

q
−
1



)



{\textstyle P(x)=\prod \left(1-f_{\alpha }(x)^{q-1}\right)}
acts as an indicator function for



V


{\displaystyle V}
. If



x
∈
V


{\displaystyle x\in V}
then



P
(
x
)
=
1


{\displaystyle P(x)=1}
, otherwise, for some




f

α




{\displaystyle f_{\alpha }}
, we have




f

α


(
x
)
≠
0


{\displaystyle f_{\alpha }(x)\neq 0}
, which implies that




f

α


(
x

)

q
−
1


=
1


{\displaystyle f_{\alpha }(x)^{q-1}=1}
, hence



P
(
x
)
=
0


{\displaystyle P(x)=0}
.
The characteristic function in convex analysis, closely related to the indicator function of a set:





χ

A


(
x
)
:=


{



0
,


x
∈
A
;




+
∞
,


x
∉
A
.








{\displaystyle \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:





φ

X


(
t
)
=
E
⁡

(

e

i
t
X


)

,


{\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}

where



E


{\displaystyle \operatorname {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.

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