# Fundamentals of the Probability Theory and the Theory of Random Processes

• V. A. Svetlitsky
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

## Abstract

In many fields of technology we have to deal with special phenomena which are usually called random phenomena. Let us consider, for example, the process of manufacturing parts of the same type. We may establish, that the dimensions of the parts will vary about a certain predetermined value. Since these deviations are of random nature, the measurements of the finished parts do not allow us to predict the dimensions of the next part. For large batches of the parts, however, dimensional deviations begin to follow certain laws, which are studied by a special mathematical discipline — the probability theory, that reflects the laws inherent in random events (phenomena) of a mass character. There are many monographs on the probability theory containing a detailed discussion of the basic concepts and methods of that theory as well as of the random functions theory. This chapter, therefore, introduces only those concepts and results related to the probability theory which have been used in the subsequent chapters of the book. One of the principal virtues of the probability theory, that enables us to use the latter effectively, for example, in the mechanical structures design, is the possibility to estimating quantitatively such emotional concepts as “probably”, “hardly probably”, “highly probable” etc. We know, that in order to design a machine, an instrument or a flying vehicle, it is necessary to obtain the numerical values of its structures parameters and of its quality (serviceability) criteria including the probability quality criteria. To compare structures according to the probability criteria we must know the numerical values of probabilities (for example, the probability of no-failure operation). The probability theory and the sections devoted to the statistical mechanics of mechanical systems, based on this theory, allow us to accomplish all of this.

## Keywords

Mathematical Expectation Numerical Characteristic Random Quantity Probability Density Distribution Random Phenomenon
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