Characteristic function (probability theory)

In probability theory, the characteristic function of a real-valued random variables defined on a probability space for the following:

It denotes the expected value. Because there exists the integral for arbitrary random variables.

Description

The characteristic function is essentially the inverse Fourier transform of the distribution of. Furthermore, the moment generating function of.

Is a real random variable with the distribution function, then applies

This results in the following two important special cases:

  • If the distribution function is absolutely continuous with density function, then
  • Is discrete with jump points, then applies

Properties

For a characteristic function is valid for any real number:

Uniform continuity

Narrowness

Symmetry

Linear transformation

Reversibility

Is integrable, then has the probability density

Torque generation

In particular, the special cases arise

If the expected value is a natural number finite, then is - times continuously differentiable in a Taylor series to be developed:

An important special case is the development of a random variable with and:

Definiteness

Each characteristic function is positive semidefinite, that is, it is for any real numbers and arbitrary complex numbers

Conversely, every positive semidefinite and uniformly continuous function with a characteristic function ( set of Bochner ).

Convolution formula for densities

For independent random variables and is valid for the characteristic function of the sum

Because true because of the independence

Uniqueness theorem

It is the following uniqueness theorem: If, random variables and applies to all, then, is that and have the same distribution function. Consequently, can thus be easily determined, the folding of some distributions.

From the uniqueness theorem, the continuity theorem of Lévy- Cramér can conclude: If a sequence of random variables, then ( convergence in distribution) if and only if for all. This property can be exploited in the central limit sets.

Examples

Discrete distributions:

  • Is a binomial distribution, then.
  • Is Poisson - distributed, then.
  • Is negative binomial distribution, then.

Absolutely continuous distributions:

  • Is a standard normal distribution, then.
  • Is normally distributed, then.
  • Is uniformly distributed, then.
  • Is gamma distributed, then.
  • Is standard Cauchy distributed, then.
Salomon Bochner Central limit theorem Cauchy distribution
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