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Sababaraha ahli matematika maké istilah fungsi karakteristik sarua jeung "fungsi indikator". The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at éach point in A and 0 at éach point that is in B but not in A.
Here t is a real number, E lambang nilai ekspektasi and F is the cumulative distribution function. The last form is valid only when f—the probability density function—exists. The form preceding it is a Riemann-Stieltjes integral and is valid regardless of whether a density function exists.
A characteristic function exists for any random variable. More than that, there is a bijection between cumulative probability functions and characteristic functions. In other words, two probability distributions never share the same characteristic function.
Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function F:
Characteristic functions are used in the most frequently seen proof of the central limit theorem.
Characteristic functions can also be used to find moments of random variable. Provided that n-th moment exists, characteristic function can be differentiated n times and
The characteristic function is closely related to the Fourier transform: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.