Cauchy distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A continuous probability distribution with density

and distribution function

where and are parameters. The Cauchy distribution is unimodal and symmetric about the point , which is its mode and median. No moments of positive order — including the expectation — exist. The characteristic function has the form . The class of Cauchy distributions is closed under linear transformations: If a random variable has the Cauchy distribution with parameters and , then the random variable also has a Cauchy distribution, with parameters and . The class of Cauchy distributions is closed under convolution:

(*)

in other words, a sum of independent random variables with Cauchy distributions is again a random variable with a Cauchy distribution. Thus, the Cauchy distribution, like the normal distribution, belongs to the class of stable distributions; to be precise: It is a symmetric stable distribution with index 1 (cf. Stable distribution). The following property of Cauchy distributions is a corollary of (*): If are independent random variables with the same Cauchy distribution, then their arithmetic mean has the same distribution as each . One more property of Cauchy distributions: In the family of Cauchy distributions, the distribution of a sum of random variables may be given by (*) even if the variables are dependent. For example, if and are independent and have the same Cauchy distribution, then the random variables and have the same Cauchy distribution. The Cauchy distribution with parameters and is the Student -distribution with one degree of freedom. The Cauchy distribution with parameters is identical with the distribution of the random variable , where and are independent and normally distributed with parameters and , respectively. A random variable with this distribution is the function , where is a random variable uniformly distributed on the interval . The Cauchy distribution is also defined in spaces of dimension greater than one. The concept was first investigated by A.L. Cauchy.

References

[1]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966)