 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-2/4f/c/solution

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

Let

U = X_{1} + X_{2}, V = X_{1} - X_{2} .

They are jointly normal because they are linear combinations of a Gaussian vector. When

σ_{1} = σ_{2} = σ

cov (U, V) = var (X_{1}) - var (X_{2}) = 0.

E U = μ_{1} + μ_{2}, var U = 2 σ^{2} (1 + ρ),

and

E V = μ_{1} - μ_{2}, var V = 2 σ^{2} (1 - ρ) .

Therefore

U \sim N (μ_{1} + μ_{2}, 2 σ^{2} (1 + ρ)),

V \sim N (μ_{1} - μ_{2}, 2 σ^{2} (1 - ρ)),

independently.

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