Let X' = [X₁,X₂] be bivariate normal with matrix of means μ' = [μ₁, μ₂] and positive definite covariance matrix Σ. Let

Show that Q₁ is χ2(r, θ) and find r and θ. When and only when does Q₁ have a central chi-square distribution?

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Q₁: X² o²(1 - p²) X1 X2 0102(1-p²) - 2p- + X2/ (1 - p²)