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The Cauchy distribution is a continuous probability distribution named after the French mathematician Augustin Cauchy. It is also known as the Lorentzian distribution. It is characterized by its "thick-tailed" shape, meaning that it has a more significant proportion of data points in the tails of the distribution than other common distributions like the normal distribution.

The Cauchy distribution has a PDF (probability density function) given by:

f(x)=1πdd2+(xm)2=1πd(1+(xmd)2)f(x) = \frac{1}{\pi}\cdot \frac{d}{d^2 + (x - m)^2} = \frac{1}{\pi d (1 + (\frac{x-m}{d})^2)}

where mm is the location parameter and dd is the scale parameter. The Cauchy distribution has no mean or variance, as the moments do not exist for this distribution. The Cauchy distribution is used in various fields, such as physics, finance, and statistics, as it is often used to model phenomena with "thick-tailed" behavior or outliers. It is also used as a test case for various statistical methods, as its heavy tails make it more challenging to estimate statistical parameters accurately.