Expectations: Applications

Abstract

Expectations are amazingly useful in the study of random variables and their distributions. Some of the reasons for this statement are contained in this chapter. In the first section, we introduce the ‘variance’ of an ℝ-valued random variable. Variance is used to obtain one version of the Law of Large Numbers, also known informally as the Law of Averages. The ‘covariance’ of two random variables is also presented. In Section 2, variance and covariance are defined for ℝ^d-valued random variables, and Section 3 concerns the expectations of various functions of ℝ-valued random variables. The chapter concludes with a discussion of ‘probability generating functions’, used in the study of distributions on Λ⁺. Several useful inequalities, including those of Chebyshev, Cauchy-Schwarz, and Jensen, are scattered throughout.