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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''.
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
::<math>S_n = \sum_{i=1}^n a_i X_i,</math>
:where the ''a'' <sub>i</sub> are constants, then the characteristic function for ''S'' <sub>n</sub> is given by
::<math>
\varphi_{S_n}(t)=\varphi_{X_1}(a_1t)\varphi_{X_2}(a_2t)\ldots \varphi_{X_n}(a_nt).
</math>
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Below was lifted from [[generating function]] ... there should be an
analog for the characteristic function
*Suppose that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function ''G''<sub>''N''</sub>. If the ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>N</sub> are independent <i>and</i> identically distributed with common probability-generating function ''G''<sub>X</sub>, then
::<math>G_{S_N}(z) = G_N(G_X(z)).</math>
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