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The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
The characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant. The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics.
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function , and one can freely convert between the two, but the characteristic function as defined below is better ...
Finally, the probability density function for is the derivative of its cumulative distribution function, which by the ... The characteristic function is given by:
The density function is therefore the inverse Fourier transform of the characteristic function: [8] = (). Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically.
The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary ...
Some writers [2] [3] prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function, [4] [5] = [] = = ()! = +