 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/29k/b/solution

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

X_{1}, \dots, X_{n}

ℓ (σ) = - n lo g σ - \frac{1}{2 σ ^{2}} \sum_{i} X_{i}^{2} .

Its unique maximum is

σ_{MLE} = \frac{1}{n} \sum_{i} X_{i}^{2} .

Now

E [X_{i}^{2}] = σ^{2}

and, using

E [Z^{4}] = 3

Var (X_{i}^{2}) = 2 σ^{4}

g (x) = x

gives

n (σ_{MLE} - σ) d N (0, σ^{2} /2) .

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