 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-39/1/ii/solution

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

Apply the law of total variance to each epoch. The Poisson distribution has equal conditional mean and variance, while

Var (T_{j}) = (2 j)^{- 2}

. Hence

Var (Y_{j}) = E [\frac{j θ T _{j}}{2}] + Var (\frac{j θ T _{j}}{2}) = \frac{θ}{j - 1} + \frac{θ ^{2}}{( j - 1 ) ^{2}} .

Var (S) = θ H_{n - 1} + θ^{2} H_{n - 1}^{(2)}, H_{n - 1}^{(2)} = i = 1 \sum n - 1 \frac{1}{i ^{2}} .

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