Bivariate Normals

Key ideas

• General normals have axis-aligned elliptical contours. Circular when $\sigma_x = \sigma_y$, stretched when they differ.
• Bivariate ($\rho \neq 0$) tilts the ellipse. The contours are no longer axis-aligned.
• Slicing at y = c gives f(x|Y=c), a normal with shifted mean $\rho\frac{\sigma_x}{\sigma_y}c$ and reduced variance $\sigma_x^2(1-\rho^2)$.