 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/16/c/solution

The tangent function is strictly increasing on the angular interval. Thus

Y \leq y

is equivalent, up to the zero-probability endpoints, to

Θ \leq arctan (y / x_{0})

. Substituting into the angular cumulative distribution function gives

F (y) = \frac{1}{2} + \frac{1}{π} arctan \frac{y}{x _{0}}, - \infty < y < \infty.

Its limits are zero and one at the two infinite endpoints. This is the construction in which the tangent of a uniform angle has a Cauchy distribution.

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