Probability theory MOC

Characteristic function

The characteristic function1 of a Real random variable is the Fourier transform of the Probability density function or equivalently the Expectation of the function 2 #m/def/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:

#state/tidy | #lang/en | #SemBr

Footnotes

  1. German charakteristische Funktion ↩

  2. 2006, Statistische Mechanik, p. 5 ↩