The Cauchy distribution
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Distribution name

Cauchy
distribution

Common notation


Parameters

=
location parameter
= scale
parameter (

Domain


Probability density
function


Cumulative distribution
function


Mean

Does not exist

Variance

Does not exist

Skewness

Does not exist

(Excess) kurtosis

Does not exist

Characteristic function


Other comments

The quantile function of the Cauchy distribution is:
Its median is thus .
The Cauchy distribution is a special case of the stable (more
precisely the sum stable) distribution family.
The special case of the Cauchy distribution when and is
called the standard Cauchy distribution. It coincides with the Student’s t
distribution with one degree of freedom. It has a probability density
function of .
If and
are
independent random variables then and this can be used to
generate random variates.
The Cauchy distribution is also known as the
CauchyLorentz or Lorentz distribution (especially amongst physicists).

Nematrian web functions
Functions relating to the above distribution may be accessed
via the Nematrian
web function library by using a DistributionName of “cauchy”. For
details of other supported probability distributions see here.
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