Man’s natural force is so well proportioned to his natural needs and his primitive state that as soon as his state changes and his needs increase ever so slightly [282] he needs his fellows’ assistance, and when eventually his desires embrace the whole of nature, the assistance of the whole of mankind barely suffices to satisfy them. That is how the same causes that make us wicked also make us slaves, and subjugate us by depraving us; the sentiment of our weakness comes less from our nature than from our cupidity: our needs unite us in proportion as our passions divide us, and the more we become our fellows’ enemies, the less can we do without them.

We introduce elementary concepts of sets, probability, and events. We then study and illustrate the basic properties of probability. We use probability to characterize independent events and mutually exclusive events. We study conditioning and Bayes' law. We also introduce essential functions required to calculate probabilities, including the factorial, gamma, and beta functions. We then apply them to calculating combinations and permutations.

This chapter is devoted to the multivariate normal and functions of it. We start by showing how linearity is essential to its definition, then we derive the main properties. These include characteristic and density functions, conditionals, and some of the normal distribution's exceptional properties: the equivalence of no-correlation and independence within the class of elliptical distributions, Cramér's deconvolution theorem, the equivalence of a random sample's normality with the independence of the sample's normal mean and chi-square variance. We also explore other properties such as fourth-order moments in multivariate normal (and elliptical) distributions, the convexity of the m.g.f., joint distributions of linear and quadratic forms and conditions for their independence, the same also for pairs of quadratic forms and their covariance, as well as decompositions of quadratic forms.

When one undertakes to write a work, one has already found the subject and at least part of the material, so that it is only a question of developing and organizing it in the way best suited to convince and to please. This part, which also includes style, is usually the part that determines the success of the work and the reputation of the Author; it is the part that makes not quite for whether a Book is good or bad, but for whether it is well or badly crafted.

The need for this chapter arises once we start considering the realistic case of more than one variate at a time, the multivariate case. We have already started dealing with this topic (in disguise) in the introductions to conditioning and mixing in Chapters 1 and 2, and in some of the exercises using these ideas in Chapter 4. Joint distributions are defined, and we explain their relation to the univariate distributions seen earlier and more generally to the distribution of subsets (marginal distributions). Joint densities are also defined. The independence of variates is defined in terms of their joint distribution. We also introduce the concept of copulas, linking the joint distribution to its marginals.