 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/29k/c/solution

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

a = Σ_{11}

c = Σ_{12}

, and

b = Σ_{22}

. The estimator is the sample mean of the independent random vectors

W_{i} = (X_{i}^{2}, X_{i} Y_{i}, Y_{i}^{2})^{T}, E W_{i} = (a, c, b)^{T} .

The supplied fourth moments give

Var (X^{2}) Var (X Y) = 2 a^{2}, = ab + c^{2}, Cov (X^{2}, X Y) Cov (X Y, Y^{2}) = 2 a c, = 2 b c, Cov (X^{2}, Y^{2}) Var (Y^{2}) = 2 c^{2}, = 2 b^{2} .

n Σ_{11} Σ_{12} Σ_{22} - a c b d N_{3} 0, 2 a^{2} 2 a c 2 c^{2} 2 a c ab + c^{2} 2 b c 2 c^{2} 2 b c 2 b^{2} .

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