characteristic function

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characteristic function

[‚kar·ik·tə′ris·tik ′fəŋk·shən]
The function χA defined for any subset A of a set by setting χA (x) = 1 if x is in A and χA = 0 if x is not in A. Also known as indicator function.
A function, such as the point characteristic function or the principal function, which is the integral of some property of an optical or mechanical system over time or over the path followed by the system, and whose value for a path actually followed by a system is a maximum or a minimum with respect to nearby paths with the same end points.
A function that uniquely defines a probability distribution; it is equal to √(2π) times the Fourier transform of the frequency function of the distribution.

Characteristic Function


in mathematics:

(1) An eigenfunction.

(2) The characteristic, or indicator, function of a set A is a function f(x) that is defined on some set E containing A and that assumes the value f(x) = 1 if x is in A and the value f(x) = 0 if x is not in A.

(3) In probability theory, the characteristic function fx(t) of a random variable X is the mathematical expectation of the quantity exp (it X). For a random variable with probability density function PX(x), this definition yields the formula

For example, for a random variable having a normal distribution with parameters a and σ, the characteristic function is

The characteristic function has several noteworthy properties. To every random variable X there corresponds a definite characteristic function. The probability distribution for X is uniquely determined by fX(t). When independent random variables are added, the corresponding characteristic functions are multiplied. If the concept of closeness is suitably defined, to random variables with close distributions there correspond characteristic functions that differ little from each other, and, conversely, to close characteristic functions there correspond random variables with close distributions. These properties underlie the applications of characteristic functions, particularly applications to the derivation of the limit theorems of probability theory.

A mathematical apparatus more or less equivalent to that of the characteristic function was first used by P. Laplace in 1812, but the full power of the characteristic function method was demonstrated in 1911 by A. M. Liapunov, who used the method in obtaining the theorem that bears his name.

The concept of characteristic function can be generalized to finite and infinite systems of random variables—that is, to random vectors and stochastic processes. The theory of characteristic functions has much in common with the theory of Fourier integrals.


Gnedenko, B. V. Kursteorii veroiatnostei, 5th ed. Moscow, 1969.
Prokhorov, Iu. V., and Iu. A. Rozanov. Teoriia veroiatnostei, 2nd ed. Moscow, 1973.

characteristic function

The characteristic function of set returns True if its argument is an element of the set and False otherwise.
References in periodicals archive ?
The corresponding characteristic function [phi]([omega]) can be denoted as:
The formula is also called the characteristic function (1), since it characterizes the performance of a particular system for a particular sample and measurand.
f] (r) known as Nevanlinna's characteristic function of f, playing the same role as maximum modulus function in the following manner:
As was the case in the preceding simulation, test characteristic functions were similar across models except at low abilities (see Fig.
Let h(t) be a positive increasing and continuous function defined on 0 < t < 1, Next, by using the above characteristic functions.
For a general cone c, we can decompose its characteristic function [c] as a signed sum of characteristic functions of unimodular cones, [[summation].
109) Professor Worthen recognizes the importance of justifying the use of the Roberts factors by identifying how they are relevant to the group's ability to serve the characteristic functions of intimate associations.
If A is a subset of an ordinary topological space then we denote its characteristic function by [x.
Our proposal is that the reduction of the bias in item parameter estimates is possible not only by increasing the number of examinees in calibration group or/and number of test items proposed to the examinees but also by expanding the set of item characteristic functions k(p) which we select the best fitting one from.
Retarded Dynamical Systems: Stability and Characteristic Functions, Vol.
i] are the same, we should compare the corresponding characteristic functions.
Thus characteristic functions of three agents have the following structure: v(Powerset(3)) = v({A}; {B}; {C}; {A;B}; {A;C};{B;C}; {A;B;C}).

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