Related Concepts
Related concepts include the moment-generating function and the probability-generating function. The characteristic function exists for all probability distributions. However this is not the case for moment generating function.
The characteristic function is closely related to the Fourier transform: the characteristic function of a probability density function p(x) is the complex conjugate of the continuous Fourier transform of p(x) (according to the usual convention; see continuous Fourier transform – other conventions).
where P(t) denotes the continuous Fourier transform of the probability density function p(x). Likewise, p(x) may be recovered from φX(t) through the inverse Fourier transform:
Indeed, even when the random variable does not have a density, the characteristic function may be seen as the Fourier transform of the measure corresponding to the random variable.
Read more about this topic: Characteristic Function (probability Theory)
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