The probability density function (pdf) of a Cauchy distribution with the location parameter a and the scale parameter b is given by
f(x|a, b) = 1 π b[1 + ((x− a)/b)2] , −∞ < a <∞, b > 0.
The cumulative distribution function (cdf) can be expressed as
F (x|a, b) = 1 2 +
(x− a b
) , b > 0. (27.1)
We refer to this distribution as Cauchy(a, b). The standard forms of the pdf and the cumulative distribution function can be obtained by replacing a with 0 and b with 1.