Cauchy distribution

2020 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]

A continuous probability distribution with density

$$ p (x; \lambda , \mu ) = \ { \frac{1} \pi } \frac \lambda {\lambda ^ {2} + (x - \mu ) ^ {2} } ,\ \ - \infty < x < \infty , $$

and distribution function

$$ F (x; \lambda , \mu ) = \ { \frac{1}{2} } + { \frac{1} \pi } \ \mathop{\rm arctan} \frac{x - \mu } \lambda , $$

where $ - \infty < \mu < \infty $ and $ \lambda > 0 $ are parameters. The Cauchy distribution is unimodal and symmetric about the point $ x = \mu $, which is its mode and median. No moments of positive order — including the expectation — exist. The characteristic function has the form $ \mathop{\rm exp} ( i \mu t - \lambda | t | ) $. The class of Cauchy distributions is closed under linear transformations: If a random variable $ X $ has the Cauchy distribution with parameters $ \lambda $ and $ \mu $, then the random variable $ Y = aX + b $ also has a Cauchy distribution, with parameters $ \lambda ^ \prime = | a | \lambda $ and $ \mu ^ \prime = a \mu + b $. The class of Cauchy distributions is closed under convolution:

$$ \tag{* } p (x; \lambda _ {1} , \mu _ {1} ) * \dots * p (x; \lambda _ {n} , \mu _ {n} ) = $$

$$ = \ p (x; \lambda _ {1} + \dots + \lambda _ {n} , \mu _ {1} + \dots + \mu _ {n} ) ; $$

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 $ X _ {1} \dots X _ {n} $ are independent random variables with the same Cauchy distribution, then their arithmetic mean $ (X _ {1} + \dots + X _ {n} ) /n $ has the same distribution as each $ X _ {k} $. 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 $ X $ and $ Y $ are independent and have the same Cauchy distribution, then the random variables $ X + X $ and $ X + Y $ have the same Cauchy distribution. The Cauchy distribution with parameters $ \lambda = 1 $ and $ \mu = 0 $ is the Student $ t $- distribution with one degree of freedom. The Cauchy distribution with parameters $ ( \lambda , \mu ) $ is identical with the distribution of the random variable $ \mu + ( X/Y ) $, where $ X $ and $ Y $ are independent and normally distributed with parameters $ (0, \lambda ^ {2} ) $ and $ (0, 1) $, respectively. A random variable with this distribution is the function $ \mu + \lambda \mathop{\rm tan} z $, where $ z $ is a random variable uniformly distributed on the interval $ [- \pi /2, \pi /2] $. The Cauchy distribution is also defined in spaces of dimension greater than one. The concept was first investigated by A.L. Cauchy.

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