Abstract
The mathematical theory of probability can be treated axiomatically in the same way as geometry. The concrete interpretations of the axioms provide a motivation for their introduction; however, they are irrelevant for the mathematical theory and have only heuristic value. The mathematical theory proceeds from the axioms in a purely deductive way, introduces new concepts by formal definitions, and derives statements concerning their properties. A real-valued, finite, and measurable function of a random variable is a random variable. This chapter presents a few elementary relations of probability theory. The characteristic functions are an important tool in the study of many problems of probability theory. The product of two characteristic functions is a characteristic function and infinitely divisible characteristic functions admit certain canonical representations. The properties of characteristic functions of random vectors are similar to those of characteristic functions of random variables. The characteristic function of the sum of a finite number of independent random variables is the product of the characteristic functions of the summands.
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