## The Bivariate Normal Distribution

The bivariate normal distribution for two related, normally distributed variablesandis

where

Ifandare independent then and

The bivariate normal PDF defines a surface in theplane.

Like its one dimensional counterpart, the bivariate normal distribution has the following properties:

and

Intuitively the probability of observing a valuebetweenandandbetweenandis

This integral is typically done numerically.