1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

Eξ = ∑ aipi.Our form of definition is however more convenient for proving thetheorems on the properties of the expectation. Let us prove, forexample, [the theorem about the expectation of a sum of variables].[...] In many textbooks that statement is proved defectively. [...]The second most important parameter of the distribution of arandom variable is variance.Definition. [...]For random variables as also for events, the concept ofindependence is most important. We define independence (in totality)for three random variables, and the definition is similar for any numberof them.Definition. [...]We will prove that for independent random variablesE(ξ ⋅ η⋅ ς) = Eξ ⋅ Eη⋅Eς.Proof. [...]It easily follows that the variance of a sum of independent randomvariables is equal to the sum of the variances of its terms.We have concluded the exposition of the main stochastic conceptsfor the discrete case, when the experiment has [only] a finite or acountable number of elementary outcomes. Now, we have to considerwhat happens when it is more natural to describe the experiment by amore complicated space.2.5. Transition to the general space of elementary events. If anexperiment results in some measurement, it is possible to state that,since the precision of all measurements is only finite, the set ofelementary outcomes will at most be countable. However, the historyof the development of science indicates that physical theories are muchsimplified by considering continuous models for which experimentalresults can be any number. Differential equations can only be appliedin such models.Readers, familiar with difference equations will easily imagine howmore elegant and simple are the differential equations. Thus, althoughmodern physics has some vague ideas about the possible discretenessof space, it certainly is not at all easy to abandon the notion ofcontinuum. And, allowing that notion, what kind of probability theoryshould we have? The answer to this question is given by thecelebrated Kolmogorov axiomatics (Kolmogorov 1933; Feller 1950and 1966). Its foundation is the notion of the space of elementaryoutcomes Ω which can now be arbitrary. Some (but not all!) of itssubspaces are held to be so to say observable as an experimental resultand called events. If A is an event, we are able to say whether itoccurred in an experiment or not and in this sense it is observable. Wemay thus discuss the frequency of its occurrence and consequently theprobability P(A).The main demand of the Kolmogorov axiomatics containing asthough in embryo the merits and shortcomings of the entire theory isthat, given a countable set of events A 1 , A 2 , ..., A n , ..., their sum and21