It occurs in physics in the approximate description of resonances , and there is called the resonance curve or Lorentz curve (after Hendrik Antoon Lorentz ). This is why there are also the names Lorentz distribution and Cauchy-Lorentz distribution .
The Cauchy distribution is a continuous probability distribution that is the probability density
with and position parameters .
The distribution function of the Cauchy distribution is
With the center and the latitude parameter , the standard Cauchy distribution (or also t-distribution with one degree of freedom ) results
as probability density and
as a distribution function.
If Cauchy distributed with the parameters and , then standard Cauchy distributed.
Expected value, variance, standard deviation, moments
The Cauchy distribution is a distribution that has neither expectation nor variance or standard deviation , since the corresponding integrals are not finite. Accordingly, it has no finite moments and no moment-generating function .
Median, mode, interquartile range
The Cauchy distribution is symmetrical to the parameter .
The entropy is .
The characteristic function of the Cauchy distribution is .
The Cauchy distribution belongs to the reproductive probability distributions: the mean value from standard Cauchy distributed random variables is itself standard Cauchy distributed. In particular, the Cauchy distribution does not obey the law of large numbers , which applies to all distributions with an existing expectation value (see Etemadi's theorem).
Invariance to convolution
The Cauchy distribution is invariant with respect to folding , that is, the convolution of a Lorentzian curve of the half-value width and a maximum at a Lorentz curve of the half-value width and a maximum at again yields a Lorentz curve with the half-width value and a maximum at . The Cauchy distribution thus forms a convolutional semigroup .
Relationships with other distributions
Relation to constant equal distribution
If the interval is continuously uniformly distributed , then the standard Cauchy distribution is used.
Relationship to normal distribution
The quotient of two independent, standard normally distributed random variables is standard Cauchy distributed.
Relationship to student t-distribution
The standard Cauchy distribution is the special case of the Student t-distribution with one degree of freedom.
Relationship to the Lévy distribution
With the Cauchy distribution as a representative of the heavy-tailed distributions , the probability of extreme characteristics is very high. If the 1% largest values of a standard normally distributed random variable are at least 2.326, the corresponding lower limit for a standard Cauchy distributed random variable is 31.82. If one wants to investigate the effect of outliers in data on statistical methods, one often uses Cauchy-distributed random numbers in simulations.
The inversion method can be used to generate Cauchy-distributed random numbers . The pseudo inverse of the distribution function to be formed according to the simulation lemma reads here (see cotangent ). For a sequence of standard random numbers, a sequence of standard Cauchy-distributed random numbers can therefore be calculated through , or, because of the symmetry, through .
- William Feller: An Introduction to Probability Theory and Its Applications: 1 . 3. Edition. Wiley & Sons, 1968, ISBN 0-471-25708-7 .
- William Feller: An Introduction to Probability Theory and Its Applications: 2 . 2nd Edition. John Wiley & Sons, 1991, ISBN 0-471-25709-5 .
- University of Konstanz - Interactive animation
- Eric W. Weisstein : Cauchy Distribution . In: MathWorld (English).