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 Fourier transform (with sign reversal) 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 multivariate probability distributions of 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 is a way to describe a random variable X.

The characteristic function,

φ

X

(

t

)

=

E

⁡

[

e

i

t

X

]

,

{\displaystyle \varphi _{X}(t)=\operatorname {E} \left[e^{itX}\right],}

a function of t,

determines the behavior and properties of the probability distribution of X.

It is equivalent to a probability density function (if it exists) or cumulative distribution function, in the sense that knowing one of these functions allows computation of the others, but they provide different insights into the features of the random variable.

In particular cases, one or another of these equivalent functions may be easier to represent in terms of simple standard functions.

If a random variable admits a density function, then the characteristic function is its Fourier dual, in the sense that each of them is a Fourier transform of the other.