Show that if (X1, X2) has the bivariate normal distribution with zero mean, variances 2 1

Show that if (X1, X2) has the bivariate normal distribution with zero mean, variances σ

2 1

, σ

2 2 and covariance ρσ1

σ2

, then the ratio U = X1

/X2 has the Cauchy distribution with median θ = ρσ1

/(σ2 and dispersion parameter τ

2 = σ

2 1 (1 − ρ

2)/σ

2 2

.

Explicitly, fU (u; θ, τ) = τ −1π

−1{1 + (u − θ)

2/τ

2}

−1

, where −∞ < θ < ∞ and τ > 0. Deduce that 1/U also has the Cauchy distribution with median θ/(τ

2 + θ

2

) and dispersion parameter τ

2

/(τ

2 + θ

2

)

2

. Interpret the conclusion that θ/τ is invariant.