Cauchy Distribution

The Cauchy distribution is a continuous probability distribution named after the French mathematician Augustin-Louis Cauchy, whose probability density forms a bell shaped curve resembling the normal distribution but with much heavier tails, so that extreme values occur far more often than the normal distribution would predict. It is the distribution followed by the ratio of two independent standard normal random variables, and it is also the distribution of the tangent of an angle chosen uniformly at random, giving it a natural geometric interpretation tied to the intersection of a fixed line with a randomly rotated line through a point. Unlike most commonly used distributions, the Cauchy distribution has no defined mean or variance, because the integrals that would ordinarily compute them fail to converge, which makes it a standard textbook counterexample to the law of large numbers, since the average of many independent Cauchy distributed samples does not settle down toward any fixed value as more samples are taken but instead remains itself Cauchy distributed.