7. Probability and Statistics Soviet Essays - Sheynin, Oscar

scientists to devote their energy, at the turn of the 19 th century, to the theory of probability;and which thus considerably fostered the development of this science.The status of probability in Europe had then been unenviable. Already in the 18 th century,the magnificent century of probability theory, Jakob Bernoulli and De Moivre discovered thetwo main laws of the doctrine of mass phenomena, the law of large numbers and the socalledlimit theorem, for the simplest particular case, – for the Bernoulli trials. The Bernoullitheorem stated that the relative frequency of any event in a given series of homogeneous andmutually independent trials should, with an overwhelming probability, be close to thatprobability which the event had at each trial. The De Moivre theorem (which had untilrecently been attributed to Laplace) stated that, under the same conditions, the probabilitiesof the values of this frequency can fairly be approximated by a formula now called thenormal distribution.Already Laplace repeatedly stated his belief in that both these theorems were valid undermuch more general conditions; in particular, he thought it highly probable that under mostgeneral conditions a sum of a very large number of independent random variables shouldpossess a distribution close to the normal law. He perceived here (and the furtherdevelopment of probability theory completely corroborated his viewpoint) the best way formathematically justifying the theory of errors of observations and measurements. Introducingthe hypotheses of elementary errors, that is, the assumption that the actual error is a sum of alarge number of mutually independent and very small as compared with this sum elementaryerrors, we may, on the strength of the abovementioned principle, easily explain the generallyknown fact that the distribution of the errors of observation is in most cases close to thenormal law. However, Laplace applied methods that did not allow to extend this principlebeyond the narrow confines indicated by the De Moivre theorem and were absolutelyinadequate for substantiating the theory of errors. Gauss is known to have chosen anotherway for attacking that goal, much less convincing in essence, but leading to it considerablyeasier 2 .In the interval between Laplace’s classical treatise and the appearance, in the second halfof the 19 th century, of the works of Chebyshev, only one bright flash, Poisson’s celebratedtreatise, had illuminated the sky of probability theory. Poisson generalized the Bernoullitheorem to events possessing differing probabilities in different trials; he called this theoremthelaw of large numbers having thus been the first who put this term into scientific circulation.There also, Poisson offered his illustrious approximation for the probabilities of seldomevents; and, finally, he made a new attempt at extending the De Moivre theorem beyond theboundaries of the Bernoulli trials. Like Laplace’s efforts, his attempt proved unsuccessful.And so, a twilight lasting all but a whole century fell over the European probability theory.Without exaggerating at all, it might be stated that, in those times, in spite of winning evermore regions of applied knowledge, European probability not only did not develop further asa mathematical science, – it literally degraded. The treatises written by Laplace and Poissonwere on a higher scientific level than the overwhelming majority of those appearing duringthe second half of the 19 th century. These latter reflect the period of decline when theencountered mathematical difficulties gradually compelled the minor scientists to follow theline of least resistance, to accept the theory of probability as a semi-empirical science only ina restricted measure demanding theoretical substantiation 3 . They usually inferred thereforethat its theorems might be proved not quire rigorously; or, to put it bluntly, that wrongconsiderations might be substituted for proofs. And, if no theoretical justification could befound for some principle, it was declared an empirically established fact. This demobilizationof theoretical thought, lasting even until now in some backward schools, has been to aconsiderable extent contributing to the compromising of the theory of probability as amathematical science. Even today, after the theory attained enormous successes during the