Theory of Probability

10Problems in Higher Dimensions

10.1 Introduction

10.1.1. It might be argued that every problem could, or even should, be put in a multidimensional framework; indeed, we have seen this over and over again throughout our treatment so far. The subject matter of this chapter is not really new, therefore, and we shall merely emphasize those features and problems which particularly relate to the multi‐dimensional nature of certain distributions.

In Chapter 6, 6.9.1, we dealt with the essential points concerning the representation of a distribution over an r‐dimensional Cartesian space, either by means of the distribution function

(10.1)

or, if it exists, by means of the density

(10.2)

In addition, we can state that a necessary and sufficient condition for a function F(x_l, x₂,…, x_r) to be a distribution function is that ƒ never be non‐negative, or, should ƒ not exist, that the expression for which it would be the limit is non‐negative. The latter is the probability of the rectangular prism given by

(10.3)

the sum being taken over the 2^r vertices corresponding to all possible ...