Cauchy distribution

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} + (xt) ^ {2}}} \ quad {\ text {for }} - \ infty

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

The distribution function of the Cauchy distribution is

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

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 {Xt} {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 .