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

For independent standard Cauchy distribution draws $X_1,\dots ,X_n$, the code averages the indicators $I_i=1_{\{X_i>2\}}$. Each is a Bernoulli random variable with parameterHence it is an unbiased estimator of that tail probability, and independence givesThe variance is finite even though the underlying Cauchy draws have no finite mean: the variables being averaged are bounded indicators, not the Cauchy values themselves.