5.2.3 The Bivariate Normal Distribution

Just as the most useful univariate distribution in statistical practice is the normal distribution, the most useful joint distribution for two rv’s $X$ and $Y$ is the bivariate normal distribution. The pdf is somewhat complicated:

$$\begin{array}{rl}f\left(x,y\right)& =\frac{1}{2\pi {\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}\exp (-\frac{1}{2(1-{\rho }^2)}[{\left(\frac{x-{\mu }_1}{{\sigma }_1}\right)}^2\\ & -2\rho \left(\frac{x-{\mu }_1}{{\sigma }_1}\right)\left(\frac{y-{\mu }_2}{{\sigma }_2}\right)+{\left(\frac{y-{\mu }_2}{{\sigma }_2}\right)}^2]).\end{array}$$

A graph of this pdf, the density surface, appears in Figure 5.6. It follows (after some tricky integration) that the marginal distribution of $X$ is normal with mean value ${\mu }_1$ and standard deviation ${\sigma }_1$, and similarly the marginal distribution of $Y$ is normal with mean ${\mu }_2$ and standard deviation ${\sigma }_2$. The fifth parameter of the distribution is $\rho$, which can be shown to be the correlation coefficient between $X$ and $Y$.

Figure 5.6 A graph of the bivariate normal pdf

It is not at all straightforward to integrate the bivariate normal pdf in order to calculate probabilities. Instead, selected software packages employ numerical integration techniques for this purpose.

EXAMPLE 5.19

It can also be shown that the conditional distribution of $Y$ given that $X=x$ is normal. This can be seen geometrically by slicing the density surface with a plane perpendicular to the $\left(x,y\right)$ passing through the value $x$ on that axis; the result is a normal curve sketched out on the slicing plane. The conditional mean value is ${\mu }_{Y\cdot x}=\left({\mu }_2-\rho {\mu }_1{\sigma }_2/{\sigma }_1\right)+\rho {\sigma }_{2}x/{\sigma }_1$, a linear function of $x$, and the conditional variance is ${\sigma }_{Y\cdot x}^2=\left(1-{\rho }^2\right){\sigma }_2^2$. The closer the correlation coefficient is to 1 or -1, the less variability there is in the conditional distribution. Analogous results hold for the conditional distribution of $X$ given that $Y=y$.

The bivariate normal distribution can be generalized to the multivariate normal distribution. Its density function is quite complicated, and the only way to write it compactly is to employ matrix notation. If a collection of variables has this distribution, then the marginal distribution of any single variable is normal, the conditional distribution of any single variable given values of the other variables is normal, the joint marginal distribution of any pair of variables is bivariate normal, and the joint marginal distribution of any subset of three or more of the variables is again multivariate normal.