High Quality Content by WIKIPEDIA articles! In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function is f(x) = {1 over pi (1 + x^2)} 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 u is any real number and o > 0, then Y = u + oX has a Cauchy distribution whose median is u and whose first and third quartiles are respectively u – o and u + o. McCullagh's parametrization, introduced by Peter McCullagh, professor of statistics at the University of Chicago uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the complex number 0 = u + io, where i is the imaginary unit. It also extends the usual range of scale parameter to include 0 < 0.

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