4.2 - Example: Bivariate Normal Distribution

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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, X1 and X2 and that these are bivariately normally distributed with mean vector components μ1 and μ2 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:

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

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 (X1, X2) for the bivariate normal distribution as shown below:

$$\phi(x_1,x_2) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\exp\{-\frac{1}{2(1-\rho^2)}\left[\left(\frac{x_1-\mu_1}{\sigma_1}\right)^2-2\rho \left(\frac{x_1-\mu_1}{\sigma_1}\right) \left(\frac{x_2-\mu_2}{\sigma_2}\right) + \left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 \right]\}$$

The links to 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 ρ 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. Open plot r = 0.0 As ρ increases that bell-shaped curve becomes flattened on the 45 degree line. So for ρ equals 0.7 we can see that the curve extends out towards minus 4 and plus 4 and becomes flattened in the perpendicular direction. Open plot r = 0.7 Increasing ρ to 0.9 the curve becomes broader and the 45 degree line and even flatter still in the perpendicular direction. Open plot r = 0.9

These three curves were produced using the SAS program normplot.sas 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.

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

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Click on the arrow in the window 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 text file phi_equation_r=0.7.txt.