Cauchy distribution

This article deals with the mathematical properties, for applications in physics see the Lorentz curve .

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 .

definition

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 . ${\displaystyle \gamma }$ ${\displaystyle x_{0}}$

The Cauchy distribution is a continuous probability distribution that is the probability density

${\displaystyle f(x)={\frac {1}{\pi }}\cdot {\frac {s}{s^{2}+(x-t)^{2}}}\quad {\text{für}}-\infty <x<\infty }$

with and position parameters . ${\displaystyle s>0}$ ${\displaystyle -\infty <t<\infty }$

The distribution function of the Cauchy distribution is

${\displaystyle F(x)=P(X\leq x)={\frac {1}{2}}+{\frac {1}{\pi }}\cdot \arctan \left({\frac {x-t}{s}}\right)}$.

With the center and the latitude parameter , the standard Cauchy distribution (or also t-distribution with one degree of freedom ) results ${\displaystyle t=0}$ ${\displaystyle s=1}$

${\displaystyle f(x)={\frac {1}{\pi (1+x^{2})}}}$

as probability density and

${\displaystyle F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\cdot \arctan(x)}$

as a distribution function.

If Cauchy distributed with the parameters and , then standard Cauchy distributed. ${\displaystyle X}$ ${\displaystyle s}$ ${\displaystyle t}$ ${\displaystyle {\frac {X-t}{s}}}$

properties

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 . ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle 2s}$

symmetry

The Cauchy distribution is symmetrical to the parameter . ${\displaystyle t}$

entropy

The entropy is . ${\displaystyle \log(4\,\pi \,s)}$

Characteristic function

The characteristic function of the Cauchy distribution is . ${\displaystyle y\mapsto \exp(ity-s|y|)}$

Reproductivity

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). ${\displaystyle (X_{1}+X_{2}+\dotsb +X_{n})/n}$ ${\displaystyle n}$

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 . ${\displaystyle \Gamma _{a}}$ ${\displaystyle t_{a}}$ ${\displaystyle \Gamma _{b}}$ ${\displaystyle t_{b}}$ ${\displaystyle \Gamma _{c}=\Gamma _{a}+\Gamma _{b}}$ ${\displaystyle t_{c}=t_{a}+t_{b}}$

Relationships with other distributions

Relation to constant equal distribution

If the interval is continuously uniformly distributed , then the standard Cauchy distribution is used. ${\displaystyle U}$ ${\displaystyle (-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}})}$ ${\displaystyle X=\tan(U)}$

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 . ${\displaystyle \alpha =1}$

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. ${\displaystyle X}$

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 . ${\displaystyle F(x)}$ ${\displaystyle F^{-1}(y)=-\cot(\pi y)}$ ${\displaystyle u_{i}}$ ${\displaystyle x_{i}:=-\cot(\pi u_{i})}$ ${\displaystyle x_{i}:=\cot(\pi u_{i})}$

literature

• 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

• University of Konstanz - Interactive animation
• Eric W. Weisstein : Cauchy Distribution . In: MathWorld (English).

See also

• Versiera of the Agnesi