7.3 Bivariate Normal distributions

Just as Normal distributions are the most important univariate distributions, joint or multivariate Normal distributions are the most important joint distributions. We mostly focus on the case of two random variables.

Jointly continuous random variables \(X\) and \(Y\) have a Bivariate Normal distribution with parameters \(\mu_X\), \(\mu_Y\), \(\sigma_X>0\), \(\sigma_Y>0\), and \(-1<\rho<1\) if the joint pdf is

{ \[\begin{align*} f_{X, Y}(x,y) & = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp\left(-\frac{1}{2(1-\rho^2)}\left[\left(\frac{x-\mu_X}{\sigma_X}\right)^2+\left(\frac{y-\mu_Y}{\sigma_Y}\right)^2-2\rho\left(\frac{x-\mu_X}{\sigma_X}\right)\left(\frac{y-\mu_Y}{\sigma_Y}\right)\right]\right), \quad -\infty <x<\infty, -\infty<y<\infty \end{align*}\] }

It can be shown that if the pair \((X, Y)\) has a BivariateNormal(\(\mu_X\), \(\mu_Y\), \(\sigma_X\), \(\sigma_Y\), \(\rho\)) distribution \[\begin{align*} \textrm{E}(X) & =\mu_X\\ \textrm{E}(Y) & =\mu_Y\\ \textrm{SD}(X) & = \sigma_X\\ \textrm{SD}(Y) & = \sigma_Y\\ \textrm{Corr}(X, Y) & = \rho \end{align*}\] A Bivariate Normal pdf is a density on \((x, y)\) pairs. The density surface looks like a mound with a peak at the point of means \((\mu_X,\mu_Y)\).

A Bivariate Normal Density has elliptical contours. For each height \(c>0\) the set \(\{(x,y): f_{X, Y}(x,y)=c\}\) is an ellipse. The density decreases as \((x, y)\) moves away from \((\mu_X, \mu_Y)\), most steeply along the minor axis of the ellipse, and least steeply along the major of the ellipse.