characteristic function - definizione. Che cos'è characteristic function
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Cosa (chi) è characteristic function - definizione

WIKIMEDIA DISAMBIGUATION PAGE
Characteristic map; Characteristic mapping; Characteristic functions

characteristic function         
<mathematics> The characteristic function of set returns True if its argument is an element of the set and False otherwise. (1995-04-13)
Characteristic function (probability theory)         
  • Pólya’s theorem can be used to construct an example of two random variables whose characteristic functions coincide over a finite interval but are different elsewhere.
FUNCTION ASSOCIATED TO A REAL-VALUED RANDOM VARIABLE THAT COMPLETELY DEFINES ITS PROBABILITY DISTRIBUTION; THE FOURIER TRANSFORM OF THE PROBABILITY DENSITY FUNCTION
Characteristic function (probability)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.
membership function         
FUNCTION THAT RETURNS 1 IF AN ELEMENT IS PRESENT IN A SPECIFIED SUBSET AND 0 IF ABSENT; NATURALLY ISOMORPHIC WITH A SET'S SUBSETS
Representing function; Characteristic sequence; Characteristic function of a set; Membership function; Indicator random variable; Indicator notation; Indicator functions

Wikipedia

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.