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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-2/26k/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 26K / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-05

The observations have finite mean and variance. The strong law of large numbers gives

\overline{X}_{n} \to E X_{1}

almost surely, so

θ_{n} \to θ

almost surely and therefore is a consistent estimator. The central limit theorem gives

n (θ_{n} - θ) d N (0, \frac{1 - α}{( 3 - α ) ( 2 - α ) ^{2}}) .

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