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:

Here t is a real number, E denotes the expected value, and f is the probability density function.

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.

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.