To further understand the multivariate normal distribution it is helpful to look at the bivariate normal distribution. Here our understanding is facilitated by being able to draw pictures of what this distribution looks like.

We have just two variables, \(X_{1}\) and \(X_{2}\) and that these are bivariately normally distributed with mean vector components \(\mu_{1}\) and \(\mu_{2}\) and variance-covariance matrix shown below:

\(\left(\begin{array}{c}X_1\\X_2 \end{array}\right) \sim N \left[\left(\begin{array}{c}\mu_1\\ \mu_2 \end{array}\right), \left(\begin{array}{cc}\sigma^2_1 & \rho \sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma^2_2 \end{array}\right)\right]\)

In this case we have the variances for the two variables on the diagonal and on the off-diagonal we have the covariance between the two variables. This covariance is equal to the correlation times the product of the two standard deviations. The determinant of the variance-covariance matrix is simply equal to the product of the variances times 1 minus the squared correlation.

\(|\Sigma| = \sigma^2_1\sigma^2_2(1-\rho^2)\)

The inverse of the variance-covariance matrix takes the form below:

\(\Sigma^{-1} = \dfrac{1}{\sigma^2_1\sigma^2_2(1-\rho^2)} \left(\begin{array}{cc}\sigma^2_2 & -\rho \sigma_1\sigma_2 \\ -\rho\sigma_1\sigma_2 & \sigma^2_1 \end{array}\right)\)

The  following three plots are plots of the bivariate distribution for the various values for the correlation row.

The first plot shows the case where the correlation \(\rho\) is equal to zero. This special case is called the circular normal distribution. Here, we have a perfectly symmetric bell-shaped curve in three dimensions.

As ρ increases that bell-shaped curve becomes flattened on the 45-degree line. So for \(\rho\) equals 0.7 we can see that the curve extends out towards minus 4 and plus 4 and becomes flattened in the perpendicular direction.

Increasing \(\rho\) to 0.9 the curve becomes broader and the 45-degree line and even flatter still in the perpendicular direction.