# Random Quantities and Random Vectors

## Abstract

By definition, values of random signals at a given sampling time are random quantities which can be distributed over a certain range of values. The tools for the precise, quantitative description of those distributions are provided by the classical *probability theory*. However natural, its development has to be handled with care since the overly heuristic approach can easily lead to apparent paradoxes. 10 But the basic intuitive idea that for independently repeated experiments, probabilities of their particular outcomes correspond to their relative frequencies of appearance, is correct. Although the concept of probability is more elementary than the concept of cumulative probability distribution functions, we assume that the reader is familiar with the former at the high school level, and start our exposition with the latter, which not only applies universally to all types of data, both discrete and continuous, but also gives us a tool to immediately introduce the probability calculus ideas, including the physically appealing probability density function.

## Keywords

Random Vector Central Limit Theorem Random Quantity Continuous Case Discrete Case## Preview

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## References

- 13.Strictly speaking, c.d.f.s that admit the integral representation (3.1.7), that is, have densities, are called
*absolutely continuous distributions*as there exist continuous c.d.f.s which do not admit this integral representation; see an example of a singular c.d.f. later in this section and, e.g., M. Denker and W. A. Woyczyński,*Introductory Statistics and Random Phenomena: Uncertainty, Complexity, and Chaotic Behavior in Engineering and Science*, Birkhäuser Boston, Cambridge, MA, 1998.MATHGoogle Scholar - 14.See, for example, M. Denker and W. A. Woyczyński,
*Introductory Statistics and Random Phenomena: Uncertainty, Complexity, and Chaotic Behavior in Engineering and Science*, Birkhäuser Boston, Cambridge, MA, 1998.MATHGoogle Scholar - 22.This error estimate in the CLT is known as the Berry-Esseen theorem and its proof can be found, for example, in V. V. Petrov’s monograph
*Sums of Independent Random Variables*, Springer-Verlag, Berlin, 1975.Google Scholar - 23.See, for example, M. Denker and W. A. Woyczyński,
*Introductory Statistics and Random Phenomena: Uncertainty, Complexity, and Chaotic Behavior in Engineering and Science*, Birkhäuser Boston, Cambridge, MA, 1998, for more details on the statistical issues discussed in this section.MATHGoogle Scholar - 25.For more information, see M. Denker and W. A. Woyczyński,