ABSTRACT
Expectations of random variables and transformed random variables are introduced in the general context of a Riemann-Stieltjes integral. In the special case of discrete or continuous probability theory, this definition is seen to reduce to the familiar notions from these theories, using Book III results. But this definition is also seen to raise existence and consistency questions. The roadmap to a final solution is outlined, foretelling needed results from the integration theory of Book V and the final detailed resolution in Book VI. Various moments, the moment generating function, and properties of such are then developed, as well as examples from the distribution functions introduced earlier. The next section then focuses on moment inequalities of Chebyshev, Jensen, Kolmogorov, Cauchy-Schwarz, Hölder, and Lyapunov, and then turns to the question of uniqueness of moments and the moment generating function. The chapter ends with an investigation of weak convergence of distributions and moment limits, developing a number of results on the “method of moments.”
