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 ?
For any two coalitions S, K [member of] [2.sup.N], the game's characteristic function satisfies the following conditions [23]:
Let [lambda] : F(P) [right arrow] [Z.sup.n.sub.2] be a characteristic function. Since [mathematical expression not reproducible] intersects at a vertex, by choosing them to be the first n-facets, one can assume that
Unlike [chi], the general characteristic function [phi] is not unique.
In Table 1, values of the characteristic function V(S) and the deviation payoffs in stage games are presented.
We split the characteristic function [[DELTA].sub.l]([lambda]) into two parts, one is known and the other is not, but it belongs to the Bernstein space [mathematical expression not reproducible], i.e.,
Additionally, it is necessary to define the characteristic functions of the system.
Precision was modeled with characteristic functions to create higher imprecision close to the limits of detection.
Then, the inverse Fourier transform will be applied to both the HDD's Fourier transform and the characteristic function of temperature [24].
Williams, "Characteristic function based estimation of stable parameters," in A Practical Guide to Heavy Tailed Data, R.
Since the probability density function of an alpha-stable random variable cannot be given in a closed-form, the characteristic function can always be given as follows:
Similarly, one can establish the following expansion for the characteristic function for the gamma- [(1-G)/G] class
We investigate critical points of Characteristic Function (CF) and calculate the behavior real and complex parts of a spectrum.

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