Cauchy distribution

The Cauchy- Lorentz distribution (after Augustin Louis Cauchy and Hendrik Antoon Lorentz ) is a continuous, leptokurtic ( supergaußförmige ) probability distribution. While the stochastic distribution is referred to as Cauchy distribution is in physics as Lorentz distribution or Lorentz - Lorentz curve or line (e.g., in the spectroscopy for the description of the shape of the spectral lines ) or a Broad- Wigner distribution (for example for the description of the resonance curve ) is known.

• 3.1 Relationship to the Lévy distribution
• 3.2 Relationship with the normal distribution
• 3.3 Relationship to Student's t-distribution

Definition

The Cauchy distribution is a continuous probability distribution, the probability density of the

And owns with.

Is the distribution function of the Cauchy distribution

With the center and the width parameter, the standard Cauchy distribution or t- distribution is obtained with a degree of freedom

Properties

Mean, variance, standard deviation, moments

The Cauchy distribution is considered to be the prototype of a distribution, which has not yet expected value variance or standard deviation, as the corresponding integrals are not defined. Similarly, they have no moments or moment generating function.

Median, mode

The Cauchy distribution has the median and mode in also included.

Entropy

The entropy is.

Characteristic function

The characteristic function of the Cauchy distribution is.

Reproductivity

The Cauchy distribution belongs to the reproductive probability distributions: the average of the standard Cauchy random variable itself is standard Cauchy distributed. In particular, the Cauchy distribution so it does not obey the law of large numbers, which applies to all distributions with existing expectation value (see set of Etemadi ).

Invariance with respect to convolution

The Cauchy distribution is invariant with respect to folding, that is, the convolution of a Lorentzian curve, the half-value width and a maximum at a Lorentz curve of the half-value width and a maximum at yields again a Lorentzian curve and the half-value width and a maximum at.

Relations with other distributions

Relationship with the Lévy distribution

The Cauchy distribution is a special α - stable Lévy distribution with exponent parameters.

Relation to the normal distribution

The quotient of two independent standard normal random variable is standard Cauchy distributed.

Relationship to Student's t-distribution

For and with the Cauchy distribution is obtained as a special case of the Student's t-distribution.

Example of use

In the Cauchy distribution, the probability of extreme characteristics is very large. Are the 1% largest values ​​of a standard normal random variables at least 2,326, as a random variable with standard Cauchy distributed is the corresponding lower limit 31,82. If you want the effect of outliers in data on statistical methods to examine is often used Cauchy- distributed random numbers in simulations.

Random numbers

To generate random numbers cauchyverteilter itself offers the inversion method. The to be formed after the Simulationslemma pseudo inverse of the cumulative distribution function is in this case (see cotangent ). To a series of standard random numbers can therefore be carried, or because of the symmetry by calculating, a consequence standard Cauchy distributed random numbers.