# McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is

${\displaystyle f(x)={1 \over \pi (1+x^{2})}}$

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 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.

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 θ = μ + iσ, where i is the imaginary unit. It also extends the usual range of scale parameter to include σ < 0.

Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters μ and σ, which can each take positive or negative values, as

${\displaystyle f(x)={1 \over \pi \left\vert \sigma \right\vert (1+{\frac {(x-\mu )^{2}}{\sigma ^{2}}})}\,,}$

where the distribution is regarded as degenerate if σ = 0. An alternative form for the density can be written using the complex parameter θ = μ + iσ as

${\displaystyle f(x)={\left\vert \Im {\theta }\right\vert \over \pi \left\vert x-\theta \right\vert ^{2}}\,,}$

where ${\displaystyle \Im {\theta }=\sigma }$.

To the question "Why introduce complex numbers when only real-valued random variables are involved?", McCullagh wrote:

In other words, if the random variable Y has a Cauchy distribution with complex parameter θ, then the random variable Y * defined above has a Cauchy distribution with parameter ( + b)/( + d).

McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at θ is the Cauchy density on the real line with parameter θ." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution".

## Differential equation

McCullagh's parametrization of the pdf of the Cauchy distribution is a solution to the following differential equation:

${\displaystyle \left\{{\begin{array}{l}f'(x)\left(\mu ^{2}+\sigma ^{2}+x^{2}-2\mu x\right)+f(x)(2x-2\mu )=0,\\f(0)={\frac {1}{\pi \left|\sigma \right|\left({\frac {\mu ^{2}}{\sigma ^{2}}}+1\right)}}\end{array}}\right\}}$