Fun with bivariate Gaussians

Suppose I give you the following bivariate Gaussian distribution,

Then I say “tell me sonny, if I drew a bunch of points from this distribution and plotted them, where about would they be?” I want a quick answer! You don’t even have time to run a Python simulation. What do you do?

Background

The mean vector is easy to understand, it’s just going to be the center of your distribution. The covariance matrix is the tricky part. Let’s recall some properties.

• First off, it’s symmetric. Why? It comes straight from the definition of covariance and the commutativity of multiplication.
• Also, it’s positive semi-definite, meaning $v^\top \Sigma v \ge 0$ for any $v$. Why? Because $v^\top \Sigma v = \text{Var}(v^\top X) \ge 0$.

Fun fact, the eigenvectors of a symmetric matrix are orthogonal!

Eigen stuff

Observe

Eigenvalue stuff