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Encyclopedia > Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

  • The most common and universal usage is as a synonym for indicator function, that is the function
mathbf{1}_A: X to {0, 1}
which for every subset A of X, has value 1 at points of A and 0 at points of X − A.
  • When applied to a natural number an effective procedure determines correctly if a natural number is or is not in the procedure's "set": "The characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set" (cf Boolos-Burgess-Jeffrey (2002) p. 73).
chi_{A} (x) := begin{cases} 0, & x in A; + infty, & x not in A. end{cases}
varphi_X(t) = operatorname{E}left(e^{itX}right),
where "E" means expected value. See characteristic function (probability theory).

  Results from FactBites:
 
Characteristic function - Wikipedia, the free encyclopedia (403 words)
The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A.
Characteristic functions are used in the most frequently seen proof of the central limit theorem.
The characteristic function is closely related to the Fourier transform: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.
  More results at FactBites »


 

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