Cauchy distribution¶
Story¶
The intercept on the xaxis of a beam of light coming from the point \((\mu, \sigma)\) is Cauchy distributed. This story is popular in physics, but is not particularly useful. You can think of it as a peaked distribution with enormously heavy tails.
Parameters¶
The Cauchy distribution is peaked, and its peak is located at \(\mu\), its location parameter, which may take on any real value. The peak’s width is dictated by a positive scale parameter \(\sigma\).
Support¶
The Cauchy distribution is supported on the set of real numbers.
Probability density function¶
Usage¶
Package 
Syntax 

NumPy 

SciPy 

Stan 

Notes¶
The
numpy.random
module only has the Standard Cauchy distribution (\(\mu=0\) and \(\sigma=1\)), but you can draw out of a Cauchy distribution using the transformation shown in the NumPy usage above.The Cauchy distribution has extremely heavy tails, so heavy that no moments are defined.