The indicator function (also called characteristic function ) is a function in mathematics that is characterized by the fact that it only takes one or two function values. It makes it possible to grasp complex sets with mathematical precision and to define functions such as the Dirichlet function on them .
There are several spellings for the characteristic function in the literature. In addition to the means used here, the spellings and are also common.
Real-valued characteristic function
A basic set and a subset are given . The function defined by
Extended characteristic function
In the optimization, the characteristic function is partly defined as an extended function . Here the function is called defined by
the characteristic function or indicator function of the set . It is a real function when it is not empty.
Partial characteristic function
Use of the different definitions
In this way, for example, case distinctions can often be avoided.
The extended characteristic function is used in optimization in order to restrict the function to sub-areas on which certain desired properties such as e.g. B. have convexity , or to model restriction sets.
The partial characteristic function is used in computability theory .
Properties and calculation rules of the real-valued characteristic function
- The amount is clearly determined by its characteristic function. It applies
- For thus follows from the equality of the equality of the quantities.
- The characteristic function of the empty set is the null function . The characteristic function of the basic set is the constant function with the value 1.
- Let there be sets . Then applies to the intersection
Use to calculate expected value, variance and covariance
and for the variance
The variance of thus assumes its maximum value in the case .
If is additional , then applies to the covariance
If there are any events, then there is the random variable
the number of those events that have occurred. Because of the linearity of the expected value, the following then applies
This formula also applies when the events are dependent. If they are also independent in pairs, then the Bienaymé equation applies to the variance
In the general case the variance can be calculated using the formula
to be determined.
- AA Konyushkov: Characteristic function of a set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Carl Geiger, Christian Kanzow: Theory and numerics of restricted optimization tasks . Springer-Verlag, Berlin Heidelberg New York 2002, ISBN 3-540-42790-2 .