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CHAPTER IThe Exponential andthe Unifortn Densities1. INTRODUCfIONIn the course of volume 1 we had repeatedly to deal with probabilitiesdefined by sums of many" small terms,. and we used approximations of theform .(1.1) P{a < X·~ b} "" f fez) tIx.The prime example is the normal appr

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significant problems do not require a clean definition of probabilities. Sometimesthey are of an analytic character and the probabilistic backgroundserves primarily as a support for our intuition. More to the point is thefact that complex stochastic processes with intricate sample spaces may leadto significant and comprehensible problems which do not depend on thedelicate tools used in the analysis of the whole process. A typical reasoningmay run as follows: if the process can be described at all, the randomvariable Z must have such and such properties, and its distribution musttherefore satisfy such and such an integral equation. Although probabilisticarguments can greatly influence the analytical treatment of the equation inquestion, the latter is in principle independent of the axioms of probability.3 Iniervals are denoted by bars to preserve the symbol (a, b) for the coordinate notationof points in the-plane. Se~ the list of abbreviations at the-front of the book.