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

For the standard Cauchy distribution tail, use the indicatorThen $\int \theta (x)f(x)dx$ is the desired probability, and the ordinary importance sampling estimator is $n^{-1}{\sum }_{i}f(x_i)1_{\{x_i\ge k\}}/g(x_i)$. The proposal must cover the nonzero integrand, namely the upper tail. A proposal restricted to $[0,k]$ cannot directly estimate this indicator integral; part (v) instead uses a complementary bounded-interval identity.