Cauchy distribution

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The Cauchy distribution (after Augustin Louis Cauchy ) is a continuous, leptokurtic (supergaussian) probability distribution .

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 .


Density function of the Cauchy distribution for different values ​​of the two parameters. The following applies: in the figure, s corresponds to the equation opposite and corresponds to  t .

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 has the median at , the mode also at , and the interquartile range .


The Cauchy distribution is symmetrical to the parameter .


The entropy is .

Characteristic function

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

The Cauchy distribution is a special α-stable distribution with the exponent parameter .

Application example

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.

Random numbers

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 .

Web links

Commons : Cauchy Distribution  - collection of images, videos, and audio files

See also