X₁,…,X₂₀ are iid N₆(μ,Σ),Σ nonsingular,μ and Σ unknown. Let X̄=ΣX_i/20 and S=Σ(X_i−X̄)(X_i−X̄)ᵀ/19. I:19S∼Wishart₆(19,Σ). II:(X₃−μ)ᵀS⁻¹(X₃−μ)∼χ²₆. Which hold?
Centering a normal sample removes one degree of freedom,so 19S is Wishart with 19 degrees of freedom,proving I. A χ²₆ quadratic requires whitening by the population covariance Σ;substituting the random S,which also depends on X₃,does not preserve that χ² law. Thus only I holds. **A**.
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