Characteristic function: Difference between revisions

Revision as of 06:19, 13 March 2005

Revision as of 23:37, 13 March 2005

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Characteristic functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example, if ''X''₁, ''X''₂, ..., ''X''_n 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'' _i are constants, then the characteristic function for ''S'' _n is given by

::<math>

\varphi_{S_n}(t)=\varphi_{X_1}(a_1t)\varphi_{X_2}(a_2t)\cdots \varphi_{X_n}(a_nt).

</math>