Basis of Probability Theory

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Abstract

We discuss the concept of a ‘random event’. The classical and statistical approaches used to formalize the notion of probability are described, along with the basic concepts of set theory and measure theory. The Kolmogorov approach for axiomatizing probability theory is presented. The probability space is introduced. The axioms of probability theory are presented, together with the addition and multiplication theorems. The notion of a scalar random variable is formalized. We present ways to describe a random variable in terms of the distribution function, probability density function, and moments, including in particular, the expectation and variance. Examples of scalar random variables with different distribution laws are presented. Methods for describing a scalar random variable are generalized to a vector random variable. The transformation of random variables and arithmetic operations on them are briefly examined.

This chapter is based on material from the books (Gorban 2003, 2016)

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Notes

1. 1.

More correctly, the smallest σ-algebra, called the Borel σ-algebra .

2. 2.

It is assumed that P(A 1) ≠ 0. Otherwise, the probability P(A 2/A 1) is not determined.

3. 3.

If it is clear from the text which random variable the distribution function concerns, the subscript on the symbol is often omitted.

4. 4.

If it is clear from the text which random variable the distribution function concerns, the subscript on the symbol is often omitted.

5. 5.

These concepts are defined in the next subsection.

6. 6.

Student is an alias of W. S. Gosset .

7. 7.

More correctly, for the Cauchy distribution, there is the principal value integral, which describes the first moment (first order moment). The value of this integral is x 0.

References

• Gorban, I.I.: Teoriya Ymovirnostey i Matematychna Statystika dla Naukovykh Pratsivnykiv ta Inzheneriv (Probability Theory and Mathematical Statistics for Scientists and Engineers). IMMSP, NAS of Ukraine, Kiev (2003)

• Gorban, I.I.: Sluchaynost i gipersluchaynost (Randomness and Hyper-randomness). Naukova Dumka, Kiev (2016)

• Gubarev, V.V.: Tablitci Kharakteristik Sluchainykh Velichin I Vektorov (Tables of Characteristics of Random Variables and Vectors). Novosibirskiy elektrotekhnicheskiy institut, Rukopis deponirovana v VINITI, 3146-81, Novosibirsk (1981)

• Gubarev, V.V.: Veroytnostnye modeli. Chast 1, 2. (Probability models. Parts 1, 2). Novosibirskiy elektrotekhnicheskiy institut, Novosibirsk (1992)

• Kolmogorov, A.N.: Obschaya teoriya mery i ischislenie veroyatnostey (General measure theory and calculation of probability). In: Proceedings of Communist Academy. Mathematics, pp. 8–21 (1929)

• Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea Publishing, New York (1956)

• Kolmogorov, A.N.: Osnovnye Ponyatiya Teorii Veroyatnostey (Fundamentals of Probability Theory). ONTI, Moscow (1974)

• Mises, R.: Grundlagen der Wahrscheinlichkeitsrechnung. Math. Z. 5, 52–99 (1919)

• Mises, R.: Mathematical Theory of Probability and Statistics. Academic, New York (1964)

• Muller, P.H., Neumann, P., Storm, R.: Tafeln der Mathematischen Statistic. VEB Fachbuchverlag, Leipzig (1979)

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Gorban, I.I. (2018). Basis of Probability Theory. In: Randomness and Hyper-randomness. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-60780-1_2