Theory of Probability

6Distributions

6.1 Introductory Remarks

6.1.1. Thus far, we have been occupied with the conceptual aspects of the formulation, and the thoroughness of the treatment reflects what the material seemed to us to require. Likewise, we have chosen to deal with the simplest topics and problems, whose meaning was not obscured by the need to involve complicated mathematics (but in fact contributed by appearing in a clear and simple light).

The time has now come, however, to abandon these self‐imposed limitations. We must examine whether, and to what extent, we can implement, in any domain whatsoever, the study of probability in terms of the image most often thought of (in an informal manner); that of the ‘distribution of mass’. In actual fact, of course, it is well known that the notion of a probability distribution (the precise mathematical translation of this image) is taken directly as the starting point in many approaches, particularly modern ones. The aim of the present chapter is to introduce this notion and the requisite mathematical tools, tying them in rigorously with our previous formulation and making any necessary modifications or limitations.

There are, therefore, two different aims to bear in mind in what follows: on the one hand, to provide a knowledge of the mathematical tools required in further study of the calculus of probability; on the other hand, to give the mathematical and conceptual details which derive from our previously established formulation and point ...