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Using Bilinear Transformations to Induce Probability Distributions

Abstract We use bilinear transformations to map points z = cos(α) + isin(α) on the unit circle in the complex plane into points x on the real line. Given any density function g(α) on the interval (−π, π), we show how a corresponding density function f(x) on (−∞, ∞) is induced. When α is uniformly distributed on (−π, π), we show that x has a Cauchy distribution in (−∞, ∞). When g(α) = Kn (1 + cos(α)) n , we show that x has a t-distribution in (−∞, ∞).