Random Variables and Distribution Functions
Section 3.1 introduces the formal definitions of random variable and its distribution, illustrated by several examples. The main properties of distribution functions, including a characterisation theorem for them, are presented in Sect. 3.2. This is followed by listing and briefly discussing the key univariate distributions. The second half of the section is devoted to considering the three types of distributions on the real line and the distributions of functions of random variables. In Sect. 3.3 multivariate random variables (random vectors) and their distributions are introduced and discussed in detail, including the two key special cases: the multinomial and the normal (Gaussian) distributions. After that, the concepts of independence of random variables and that of classes of events are considered in Sect. 3.4, establishing criteria for independence of random variables of different types. The theorem on independence of sigma-algebras generated by independent algebras of events is proved with the help of the probability approximation theorem. Then the relationships between the introduced notions are extensively discussed. In Sect. 3.5, the problem of existence of infinite sequences of random variables is solved with the help of Kolmogorov’s theorem on families of consistent distributions, which is proved in Appendix 2. Section 3.6 is devoted to discussing the concept of integral in the context of Probability Theory (a formal introduction to Integration Theory is presented in Appendix 3). The integrals of functions of random vectors are discussed, including the derivation of the convolution formulae for sums of independent random variables.
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