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# Characteristic function

Some mathematicians use the phrase characteristic function 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.

In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

$\varphi(t)=E\left(e^{itX}\right).$
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.

Related concepts include the moment-generating function and the probability-generating function.

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