# 4.8 - Special Cases: p = 2

4.8 - Special Cases: p = 2

To further understand the shape of the multivariate normal distribution, let's return to the special case where we have p = 2 variables.

If $$ρ = 0$$, there is zero correlation, and the eigenvalues turn out to be equal to the variances of the two variables. So, for example, the first eigenvalue would be equal to $$\sigma^2_1$$ and the second eigenvalue would be equal to $$\sigma^2_2$$  as shown below:

$$\lambda_1 = \sigma^2_1$$ and $$\lambda_2 = \sigma^2_2$$

the corresponding eigenvectors will have elements 1 and 0 for the first eigenvalue and 0 and 1 for the second eigenvalue.

$$\mathbf{e}_1 = \left(\begin{array}{c} 1\\ 0\end{array}\right)$$, $$\mathbf{e}_2 = \left(\begin{array}{c} 0\\ 1\end{array}\right)$$

So, the axis of the ellipse, in this case, are parallel to the coordinate axis.

If there is zero correlation, and the variances are equal so that $$\sigma^2_1$$ = $$\sigma^2_2$$, then the eigenvalues will be equal to one another, and instead of an ellipse we will get a circle. In this special case, we have a so-called circular normal distribution.

If the correlation is greater than zero, then the longer axis of the ellipse will have a positive slope.

Conversely, if the correlation is less than zero, then the longer axis of the ellipse will have a negative slope.

As the correlation approaches plus or minus 1, the larger eigenvalue will approach the sum of the two variances, and the smaller eigenvalue will approach zero:

$$\lambda_1 \rightarrow \sigma^2_1 +\sigma^2_2$$ and $$\lambda_2 \rightarrow 0$$

So, what is going to happen in this case is that the ellipse becomes more and more elongated as the correlation approaches one.

## Using Technology

#### Using SAS

The SAS program below can be used to plot the 95% confidence ellipse corresponding to any specified variance-covariance matrix.