7. Probability and Statistics Soviet Essays - Sheynin, Oscar

period 1 and remained, until the appearance of De Moivre’s work, the only limit theorem ofthe theory of probability.4. In the next, the second period according to my reckoning, separate fields had alreadyappeared where quantitative probability-theoretic calculations were required. These fieldswere not yet numerous. The main spheres of application were the theory of errors andproblems in the theory of artillery firing. The chief results obtained in the former theory wereconnected with Gauss, and the achievements in the latter subject, with Poisson 2 . Neitherfield was, however, alien for Laplace who was the main figure of that time. Here are the mainpertinent theoretical results.1) The De Moivre – Laplace limit theorem. It asymptotically estimates the probabilityP n (t) = P(µ ≤ np + t np( 1−p)that, in n independent trials, each having probability p of a positive outcome, the number ofsuch outcomes µ will not exceed np + t np( 1− p). The theorem states that, as n , P n (t)tends toP(t) = (1/t2 π ) −∞exp(– x 2 /2)dx.From then onwards, the probability distribution P(t), appearing here for the first time, isplaying a large part in the entire further theory of probability and is {now} called normal.2) The Poisson generalization of this theorem to the case of variable probabilities p 1 , p 2 , …,p n .3) The substantiation of the method of arithmetic mean {of least squares} by Gauss.4) The development of the method of characteristic functions by Laplace.Thus, not only from the ideological and philosophical side, but in the regular everydayscientific work, the main attention was transferred from the elementary theorems about afinite number of events to limit theorems. Accordingly, non-elementary analytic methodswere dominating.Note that the maturity of the contemporary Russian science revealed itself in thatLobachevsky’s probability-theoretic work, in spite of his remote peripheral scientific interestin the theory of probability, was quite on a level with international science and approvinglyquoted by Gauss 3 . Ostrogradsky also left several works in probability, but the dominantinfluence of Russian science on the entire development of probability theory begins later.5. The third period in the development of the theory of probability, i.e., the second half ofthe 19 th century, is especially interesting for us. In those times, a rapid development ofmathematical statistics and statistical physics occurred in Western Europe. However, it tookplace on a rather primitive and dated theoretical basis with Petersburg becoming the center ofstudies in the main general problems of probability. The activity of academicianBuniakovsky, who, in 1846, published an excellent for his time treatise, ! (Principles of the Mathematical Theory ofProbability), and widely cultivated applications of probability to insurance, statistics, and,especially, demography, and paved the way for the flourishing of the Petersburg school ofprobability theory.It was Pafnuty Lvovich Chebyshev, however, who brought the Russian theory ofprobability to the first place in the world. From a methodological aspect, the principalupheaval accomplished by him consisted not only in that he was the first to demand, with