ABSTRACT
From now on, (Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203742013/69a53f4e-97e6-48d7-bf94-69ab32e54fc6/content/f.tif"/>, P) will denote a probability space. If X is a random variable, for instance, if X is the outcome of a random experiment, then the expectation EX should be that real value which we expect before the experiment has been performed; that is, EX should be that real number which in some sense is as close as possible to the many values that our random variable may take. The first textbook on probability theory (from 1657) was written by Christaan Huyghens (1629-1695) entitled De Ratiociniis in Ludo Alea which means “How to reason in the game of dice.” In his book, Huyghens take the expectation as the fundamental concept of probability, and probabilities are only expressed in terms of odds. For instance, “To have p chances of winning a and q chances of winning b,” means in modern terminology that the probability of winning a equals p p + q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203742013/69a53f4e-97e6-48d7-bf94-69ab32e54fc6/content/in249_2.tif"/> and the probability of winning b equals q p + q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203742013/69a53f4e-97e6-48d7-bf94-69ab32e54fc6/content/in249_3.tif"/> . In his three first theorems, Huyghens defines the expectation (in his terminology: “The value of my chance”) as follows:
