Characteristic Function (probability Theory)
In probability theory and statistics, 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 Inverse Fourier transform of the probability density function. Thus it provides the basis of 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 even 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.
Read more about Characteristic Function (probability Theory): Introduction, Definition, Generalizations, Examples, Properties, Uses, Entire Characteristic Functions, Related Concepts
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