Probability Theory, 2nd Edition

6 Normally distributed random vectors

6.1 Representation and density

In Example 3.4.3 we considered a two-dimensional random vector $\mathrm{(}{\mathit{X}}_1\mathrm{,}{\mathit{X}}_2\mathrm{)}$, where ${\mathit{X}}_1$ was the height of a randomly chosen person and ${\mathit{X}}_2$ was his weight. From experience and in view of the central limit theorem (cf. Section 7.2), it is quite reasonable to assume that ${\mathit{X}}_1$ and ${\mathit{X}}_2$ are normally distributed. Suppose we are able to determine their expected values and their variances. However, this is not sufficient to describe the experiment. Why? The random variables ${\mathit{X}}_1$ and ${\mathit{X}}_2$ are surely dependent, and the most interesting problem is to describe their degree of dependence. This cannot be done based only on the knowledge of their distributions. What we really need to know is their joint ...