Characteristic function

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

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 let be their vanishing locus. Then, the function acts as an indicator function for . If then , otherwise, for some , we have , which implies that , hence .
  • The characteristic function in convex analysis, closely related to the indicator function of a set:
  • 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:
where E means expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.


  1. ^ Serre. Course in Arithmetic. p. 5.