Final Answer:
The tangent of a random angle uniformly distributed on [0, 2π) has a Cauchy distribution C(0, 1).
Explanation:
The Cauchy distribution arises when considering the tangent of a uniform random angle distributed on [0, 2π). The tangent function relates to sine and cosine, and its behavior near the vertical asymptotes leads to the Cauchy distribution. When taking the tangent of an angle uniformly distributed between 0 and 2π, the density function of the resulting distribution follows the Cauchy distribution with a location parameter of 0 and a scale parameter of 1.
The Cauchy distribution is characterized by its fat tails and lack of finite moments. It has a heavy tail, implying that extreme values occur more frequently compared to other distributions with finite variance. As the tangent function has vertical asymptotes at regular intervals (π/2 apart), these asymptotes influence the distribution's properties, resulting in a Cauchy distribution when applied to a uniformly distributed angle.
The peculiar nature of the Cauchy distribution, with its lack of moments and heavy tails, emerges due to the way the tangent function operates on uniformly distributed angles. This behavior leads to the observation that the tangent of a random angle uniformly distributed on [0, 2π) follows a Cauchy distribution with parameters (0, 1).