In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the characteristic function of any
real-valued random variable completely defines its
probability distribution. If a random variable admits a
probability density function, then the characteristic function is the
Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with
probability density functions or
cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
In addition to
univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases.
The characteristic function always exists when treated as a function of a real-valued argument, unlike the
moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.
Introduction
The characteristic function provides an alternative way for describing a
random variable. Similar to the
cumulative distribution function,
: