# characteristic function

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## characteristic function

[‚kar·ik·tə′ris·tik ′fəŋk·shən]
(mathematics)
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.
(physics)
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.
(statistics)
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.

### REFERENCES

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

## characteristic function

(mathematics)
The characteristic function of set returns True if its argument is an element of the set and False otherwise.
References in periodicals archive ?
Take the logarithm for formula (1) (2), and the second order characteristic functions of x, [bar.
The value of this force should be such determined that the movement of this system would be consistent with the requirements assumed with regard to the characteristic function.
This means that a possible lack of ability of the characteristic function to model the data was of a magnitude that was equal to or smaller than the rounding error of the reported results.
But, again, test characteristic functions are very similar across models except at low abilities (see Fig.
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173) Applying functional analysis, however, a court would have to use those and any other relevant factors to support its explicit conclusion that the group had failed to fulfill a characteristic function of intimate associations.
In this section we give a brief subsection of some concepts and notation related to bankruptcy problem, as well as preliminaries related to n-person cooperative games in characteristic function form, since it is a key subject in subsequent developments.
For any characteristic function, Shapley (1953) showed that there is a unique reward vector.
That is if X follows ETL(Q) distribution with characteristic function (2.
3x3] the characteristic matrix of the 2-person coalitions and w a 3-vector with characteristic function values w = [([v.
i]: D(S) [right arrow] {0, 1} be the characteristic function of the set [D.

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