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.
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
For a characteristic function is valid for any real number:
Is integrable, then has the probability density
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:
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
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.
- 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.