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
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- 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.
Famous quotes containing the word function:
“Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the implications and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.”
—Susanne K. Langer (1895–1985)