Probability theory MOC

Characteristic function

The characteristic function^[German charakteristische Funktion] of a Real random variable is the Fourier transform of the Probability density function or equivalently the Expectation of the function ¹ prob

which is a complex analogue to the moment-generating function. This describes the distribution of completely — the density function may be obtained using the reverse Fourier transform:

Using the Taylor series expansion of one obtains a further representation of in terms of moments:

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tidy | en | sembr

Footnotes

1. 2006, Statistische Mechanik, p. 5 ↩