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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''. In fact, the probability distribution function is equal to the Fourier transform of the characteristic function (up to a constant of proportionality and assuming the integral is defined)
:<math>f_X(x) = \frac{1} {2\pi} \int_{-\infty}^{+\infty} e^{-itx}\, \phi_X(t)\, dt.</math>
Characteristic functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example, if ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> is a sequence of independent (and not necessarily identically distributed) random variables, and
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