McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is for x real. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if X has a standard Cauchy distribution and μ is any real number and σ > 0, then Y = μ + σX has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ. where .

McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is for x real. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if X has a standard Cauchy distribution and μ is any real number and σ > 0, then Y = μ + σX has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ. where .