Follow

Assume that (Y₁, Y₂) follows a bivariate distribution with mean vector (μ₁, μ₂) and covariance matrix with entries σ₁², σ₂², σ₁₂, and let ρ=Cor(Y₁,Y₂).

If we collect i.i.d. observations (Yᵢ₁, Yᵢ₂) from this bivariate distribution, then the expected squared perpendicular distance of each such point from the 45 degree line in 2d plane is given by

E[(Y₁-Y₂)²] =
E[((Y₁-μ₁) - (Y₂-μ₂) + μ₁-μ₂)²] =
E[((Y₁-μ₁) - (Y₂-μ₂))²] + (μ₁-μ₂)² =
(μ₁-μ₂)²+σ₁²+σ₂²-2σ₁₂ =
(μ₁-μ₂)²+(σ₁-σ₂)²+2(1-ρ)σ₁σ₂

Sign in to participate in the conversation
Mathstodon

A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes. Use \( and \) for inline LaTeX, and \[ and \] for display mode.