Characteristic function: Difference between revisions

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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>

If ''X'' is a [[vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].

A related concept is the idea of a [[moment-generating function]].