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Some mathematicians use the phrase '''<i>characteristic function</i>''' synonymously with "[[indicator function]]". The indicator function of a [[subset]] ''A'' of a [[set]] ''B'' is the [[function]] with ___domain ''B'', whose value is 1 at each point in ''A'' and 0 at each point that is in ''B'' but not in ''A''.
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In [[probability theory]], the '''characteristic function''' of any [[probability distribution]] on the [[real number|real]] line is given by the following formula, where ''X'' is any [[random variable]] with the distribution in question:
:<math>\varphi(t)=E\left(e^{itX}\right).</math>
Here ''t'' is a [[real number]] and E denotes the [[expected value]].
If ''X'' is a [[vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].
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