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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-40/4/a/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

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 parameter

θ = \int_{2}^{\infty} \frac{d x}{π ( 1 + x ^{2} )} = \frac{1}{2} - \frac{a r c t a n 2}{π} = \frac{a r c t a n ( 1/2 )}{π} .

(1)

Hence it is an unbiased estimator of that tail probability, and independence gives

θ = \frac{1}{n} i = 1 \sum n I_{i}, θ \approx 0.1475836, Var (θ) = \frac{θ ( 1 - θ )}{n} .

(2)

The 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.

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