Characteristic Function

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

  • The most common and universal usage is as a synonym for indicator function, that is the function
which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
  • 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. This concept extends to multivariate distributions.
  • The characteristic function in convex analysis:
  • The characteristic state function in statistical mechanics.
  • The characteristic polynomial in linear algebra.
  • The Euler characteristic, a topological invariant.
  • The cooperative game in game theory.

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