Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 40 / 4 / b / Solution

For $Y=2/(1-U)$ with $U$ uniform on $(0,1)$,Thus $Y$ has a Pareto distribution and is the importance sampling proposal, concentrated on the required tail. The target tail integrand is $f(y)=1/(\pi (1+y^2))$, so its importance weight isThe function returns $\tilde{\theta }=n^{-1}{\sum }_{i}w(Y_i)$. Since ${\mathbb{E}}_{q}w(Y)={\int }_2^{\infty }f(y)dy=\theta$, it is an unbiased estimator. For its second moment,An antiderivative is $\frac{1}{2}(\arctan y-y/(1+y^2))$, so the integral is $\frac{1}{2}(\pi \theta +2/5)$. Independence of the proposals givesThis is importance sampling of a Cauchy tail, with weights between $2/(5\pi )$ and $1/(2\pi )$. Their narrow range explains the much smaller variance than the indicator estimate, without appealing to nonexistent Cauchy moments.