Suppose that (1, 2, X3) have the trivariate normal distribution with zero mean and intra-class covariance matrix

Suppose that (Χ1, Χ2, X3) have the trivariate normal distribution with zero mean and intra-class covariance matrix with variances σ

2 and correlations ρ. Show that

−

1 2 ≤ ρ ≤ 1. Prove that the moments of Χ1 + ωΧ2 + ω

2Χ3 and X1 + ωΧ3 + ω

2Χ2 are independent of both parameters, but that neither statistic is ancillary [ω = exp(2πi/3)].