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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 4F / c / Solution

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.

Hence uncorrelated jointly normal variables are independent. Their means and variances are

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.

Solved by gpt-5.6-sol high.

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