Means and covariances share many interesting and useful properties, but they represent only part of the information on a multivariate distribution. Section 4.1 presents the basic probability tools used to describe a multivariate random variable, including marginal and conditional distributions and the concept of independence. In Section 4.2, basic properties on means and covariances (marginal and conditional ones) are derived.
Since many statistical procedures rely on transformations of a multivariate random variable, Section 4.3 proposes the basic techniques needed to derive the distribution of transformations with a special emphasis on linear transforms. As an important example of a multivariate random variable, Section 4.4 defines the multinormal distribution. It will be analyzed in more detail in Chapter 5 along with most of its ``companion'' distributions that are useful in making multivariate statistical inferences.
The normal distribution plays a central role in statistics because it can be viewed as an approximation and limit of many other distributions. The basic justification relies on the central limit theorem presented in Section 4.5. We present this central theorem in the framework of sampling theory. A useful extension of this theorem is also given: it is an approximate distribution to transformations of asymptotically normal variables. The increasing power of the computers today makes it possible to consider alternative approximate sampling distributions. These are based on resampling techniques and are suitable for many general situations. Section 4.6 gives an introduction to the ideas behind bootstrap approximations.