Numerical Characteristics of Random Variables
This chapter opens with Sect. 4.1 introducing the concept of the expectation of random variable as the respective Lebesgue integral and deriving its key properties, illustrated by a number of examples. Then the concepts of conditional distribution functions and conditional expectations given an event are presented and discussed in detail in Sect. 4.2, one of the illustrations introducing the ruin problem for the simple random walk. In the Sects. 4.3 and 4.4, expectations of independent random variables and those of sums of random numbers of random variables are considered. In Sect. 4.5, Kolmogorov–Prokhorov’s theorem is proved for the case when the number of random terms in the sum is independent of the future, followed by the derivation of Wald’s identity. After that, moments of higher orders are introduced and discussed, starting with the variance in Sect. 4.5 and proceeding to covariance and correlation coefficient and their key properties in Sect. 4.6. Section 4.7 is devoted to the fundamental moment inequalities: Cauchy–Bunjakovsky’s inequality (a.k.a. Cauchy–Schwarz inequality), Hölder’s and Jensen’s inequalities, followed by inequalities for probabilities (Markov’s and Chebyshev’s inequalities). Section 4.8 extends the concept of conditional expectation (given a random variable or sigma-algebra), starting with the discrete case, then turning to square-integrable random variables and using projections, and finally considering the general case basing on the Radon–Nykodim theorem (proved in Appendix 3). The properties of the conditional expectation are studied, following by introducing the concept of conditional distribution given a random variable and illustrating it by several examples in Sect. 4.9.
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