4.2 - Bivariate Normal Distribution

Bivariate Normal Distribution Section

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:

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

Joint Probability Density Function for Bivariate Normal Distribution
Substituting in the expressions for the determinant and the inverse of the variance-covariance matrix we obtain, after some simplification, the joint probability density function of (\(X_{1}\), \(X_{2}\)) for the bivariate normal distribution as shown below:
\(\phi(x_1,x_2) = \dfrac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\exp\{-\dfrac{1}{2(1-\rho^2)}\left[\left(\dfrac{x_1-\mu_1}{\sigma_1}\right)^2-2\rho \left(\dfrac{x_1-\mu_1}{\sigma_1}\right) \left(\dfrac{x_2-\mu_2}{\sigma_2}\right) + \left(\dfrac{x_2-\mu_2}{\sigma_2}\right)^2 \right]\}\)

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

Plot when r is qual to 0

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.

Plot when r is equal to 0.7

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.

plot when r is equal to 0.9

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

Using Technology Section

Using SAS

These three curves were produced using the SAS program shown below. The desired correlation is specified in the third line of the SAS code (here at 0.9). No other changes are required to run this program. It would be a good idea to try this program for various values of r between -1 and 1 to explore how the shape of the normal distribution varies with the correlation.

Download the Normal plot SAS program here normplot.sas

Note! This code assumes that the variances are both equal to one.

Using Minitab

View the video below to see how you can use Minitab to create plots of the bivariate distribution.
You will need the formula that is found in the downloadable text file here: phi_equation_r=0.7.txt.