Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 4 / v / Solution

Take $k>0$, so the proposal is the uniform distribution on $[0,k]$, with probability density function $g(x)=1/k$. Symmetry of the standard Cauchy distribution givesThus the proposed expression is exactly the unbiased importance sampling estimator of this complementary integral:It has finite variance because $f$ is bounded on $[0,k]$. Explicitly,These follow from the antiderivatives of $(1+x^2{)}^{-1}$ and $(1+x^2{)}^{-2}$. The target itself is $\mu =1/2-\arctan (k)/\pi$. At $k=0$ it is exactly $1/2$ without simulation; for negative $k$ the printed uniform-interval construction must be changed.