 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/28k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Since

\overline{X}_{n} \sim N (μ, 1/ n)

E T_{n} = E \overline{X}_{n}^{2} = μ^{2} + \frac{1}{n},

Bias (T_{n}) = \frac{1}{n} .

Every leave-one-out mean has variance

1/ (n - 1)

, and hence

E B_{n} = (n - 1) (\frac{1}{n - 1} - \frac{1}{n}) = \frac{1}{n} .

Therefore

Bias (T_{JACK}) = 0 .

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