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

Nowadays we are sure that no independence of the judgement ofindividual jurors does exist, so that the groups isolated by Cournotwould have most likely consisted of one case only. True, thisstatement is not really proven so that according to modern scienceCournot’s point of view is formally invulnerable which once againconfirms that he had essentially outstripped his time.For our days, an important conclusion from the above is that it is byno means permissible to use all the available statistical information fordetermining the parameters of a statistical model; it is absolutelynecessary to leave some part of it for checking the model itself,otherwise, great scientific efforts can result in complete rubbish.4. Substantial Theorems of the Theory of Probability4.1. The Poisson theorem. When compiling his treatise, Poisson(1837) discovered one of the main statistical laws. Calculating theprobabilities P(µ = m) that m successes will be achieved in n Bernoullitrials, he found out an approximate formula for large values of n andsmall values of p:λP{µ m}em!m−λ= ≈ (4.1)where λ = np; for more details see Gnedenko (1950). The exactexpression for P{µ = m} depends on three parameters, n, m and p; inthe approximate expression, n and p are combined into one.At first sight this simplification seems trivial and Poisson himselfdid not think that his formula was really important. Indeed, his treatiseincluded a large number of more precise and almost as suitableformulas. However, the combining mentioned allows to compile acomparatively short table for calculating (4.1) with two entries, m andλ, whereas the precise expression for P{µ = m} would have demandeda table with three entries which is not done yet in a sufficientlyconvenient form.Nevertheless, the main role of the formula (4.1) consists not inconvenient calculation. Strictly speaking, we express it as amathematical theorem (Gnedenko 1950) concerning Bernoulli trials, i.e. independent trials with two outcomes and a constant probability ofsuccess. The most important circumstance is that those conditions maybe violated without denying its conclusion, that is, the equality (4.1).For example, we may admit that different trials have differingprobabilities of successp 1 , p 2 , ..., p n , ...(if only all of them are low). Then the exact expression for P{µ = m}from Chapter 3 as well as good enough approximate expressionsbecome useless (because they are too exact). The comparatively roughexpression (4.1) remains valid if only p 1 + p 2 + ...+ p n , or, if desired,np be substituted instead of λ. It follows that for calculating λ it is notnecessary to know the values of p i , suffice it to know one singleparameter, their mean, the new value of the probability of success.29