Q 8.4. Let X₁, X2, Y₁ and Y₂ be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] = ₁, that E[Y₁] = E[Y₂] = µ2 and that var(X₁) = var(X₂) = var(Y₁) = var(Y₂) = 0² > 0. Let X = (X₁ + X₂) and Ỹ = (Y₁ + Y₂). Define a random variable S² via 82 - (2 = i=1 The aim of this question is to describe the joint distribution of X - Y and S² 2 Σ(Xi − X)² + Σ(Y₁ – Ý)² + Σm-n²) i=1

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