Take the <indicator function> $\theta(x)=\mathbf1_{\{x\geq k\}}$. Its <expectation> under the standard <Cauchy distribution> is the desired upper-tail <probability>. To use the proposed tail <probability density function> in the remaining parts, \b[assume $k>0$]: a <probability density function> proportional to $x^{-2}$ on $[k,\infty)$ is not normalizable when that interval contains zero.

Solution

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 40 / 2 / b / i / Solution

 Solution (source code)