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:

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*.